LMFDB - Embedded newform 4028.2.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, 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("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.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:	$32.1637419342$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 124 x^{16} + 364 x^{15} - 1554 x^{14} - 2310 x^{13} + 10113 x^{12} + \cdots + 139 $ x^19 - 4x^18 - 30x^17 + 124x^16 + 364x^15 - 1554x^14 - 2310x^13 + 10113x^12 + 8368x^11 - 36567x^10 - 18074x^9 + 72868x^8 + 23819x^7 - 73816x^6 - 17877x^5 + 30973x^4 + 5074x^3 - 4299x^2 - 454x + 139
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.30353$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30353 q^{3} +0.631218 q^{5} -0.267329 q^{7} +2.30625 q^{9} +O(q^{10})$	$q+2.30353 q^{3} +0.631218 q^{5} -0.267329 q^{7} +2.30625 q^{9} +3.34594 q^{11} +5.71167 q^{13} +1.45403 q^{15} +3.40139 q^{17} -1.00000 q^{19} -0.615800 q^{21} +3.34091 q^{23} -4.60156 q^{25} -1.59807 q^{27} +0.469687 q^{29} -3.82667 q^{31} +7.70748 q^{33} -0.168743 q^{35} -6.35693 q^{37} +13.1570 q^{39} +11.1219 q^{41} +6.63458 q^{43} +1.45575 q^{45} +3.70119 q^{47} -6.92854 q^{49} +7.83520 q^{51} -1.00000 q^{53} +2.11202 q^{55} -2.30353 q^{57} +0.818467 q^{59} +1.27038 q^{61} -0.616528 q^{63} +3.60531 q^{65} -9.76418 q^{67} +7.69590 q^{69} -4.93177 q^{71} -9.25090 q^{73} -10.5998 q^{75} -0.894467 q^{77} +4.83133 q^{79} -10.6000 q^{81} +7.98855 q^{83} +2.14702 q^{85} +1.08194 q^{87} -14.4971 q^{89} -1.52689 q^{91} -8.81484 q^{93} -0.631218 q^{95} +12.1173 q^{97} +7.71659 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	$ 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) $ 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * 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
$3$	2.30353	1.32994	0.664972	−	0.746868i	$-0.268443\pi$
$3$	2.30353	1.32994	0.664972	+	0.746868i	$0.268443\pi$
$4$	0	0
$5$	0.631218	0.282289	0.141145	−	0.989989i	$-0.454922\pi$
$5$	0.631218	0.282289	0.141145	+	0.989989i	$0.454922\pi$
$6$	0	0
$7$	−0.267329	−0.101041	−0.0505204	−	0.998723i	$-0.516088\pi$
$7$	−0.267329	−0.101041	−0.0505204	+	0.998723i	$0.516088\pi$
$8$	0	0
$9$	2.30625	0.768751
$10$	0	0
$11$	3.34594	1.00884	0.504420	−	0.863459i	$-0.331706\pi$
$11$	3.34594	1.00884	0.504420	+	0.863459i	$0.331706\pi$
$12$	0	0
$13$	5.71167	1.58413	0.792066	−	0.610435i	$-0.209006\pi$
$13$	5.71167	1.58413	0.792066	+	0.610435i	$0.209006\pi$
$14$	0	0
$15$	1.45403	0.375429
$16$	0	0
$17$	3.40139	0.824957	0.412479	−	0.910967i	$-0.364663\pi$
$17$	3.40139	0.824957	0.412479	+	0.910967i	$0.364663\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.615800	−0.134379
$22$	0	0
$23$	3.34091	0.696629	0.348314	−	0.937378i	$-0.386754\pi$
$23$	3.34091	0.696629	0.348314	+	0.937378i	$0.386754\pi$
$24$	0	0
$25$	−4.60156	−0.920313
$26$	0	0
$27$	−1.59807	−0.307548
$28$	0	0
$29$	0.469687	0.0872187	0.0436093	−	0.999049i	$-0.486114\pi$
$29$	0.469687	0.0872187	0.0436093	+	0.999049i	$0.486114\pi$
$30$	0	0
$31$	−3.82667	−0.687289	−0.343645	−	0.939100i	$-0.611662\pi$
$31$	−3.82667	−0.687289	−0.343645	+	0.939100i	$0.611662\pi$
$32$	0	0
$33$	7.70748	1.34170
$34$	0	0
$35$	−0.168743	−0.0285227
$36$	0	0
$37$	−6.35693	−1.04507	−0.522537	−	0.852617i	$-0.675014\pi$
$37$	−6.35693	−1.04507	−0.522537	+	0.852617i	$0.675014\pi$
$38$	0	0
$39$	13.1570	2.10681
$40$	0	0
$41$	11.1219	1.73694	0.868472	−	0.495738i	$-0.165102\pi$
$41$	11.1219	1.73694	0.868472	+	0.495738i	$0.165102\pi$
$42$	0	0
$43$	6.63458	1.01176	0.505882	−	0.862603i	$-0.331167\pi$
$43$	6.63458	1.01176	0.505882	+	0.862603i	$0.331167\pi$
$44$	0	0
$45$	1.45575	0.217010
$46$	0	0
$47$	3.70119	0.539873	0.269937	−	0.962878i	$-0.412997\pi$
$47$	3.70119	0.539873	0.269937	+	0.962878i	$0.412997\pi$
$48$	0	0
$49$	−6.92854	−0.989791
$50$	0	0
$51$	7.83520	1.09715
$52$	0	0
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	0	0
$55$	2.11202	0.284785
$56$	0	0
$57$	−2.30353	−0.305110
$58$	0	0
$59$	0.818467	0.106555	0.0532776	−	0.998580i	$-0.483033\pi$
$59$	0.818467	0.106555	0.0532776	+	0.998580i	$0.483033\pi$
$60$	0	0
$61$	1.27038	0.162656	0.0813278	−	0.996687i	$-0.474084\pi$
$61$	1.27038	0.162656	0.0813278	+	0.996687i	$0.474084\pi$
$62$	0	0
$63$	−0.616528	−0.0776752
$64$	0	0
$65$	3.60531	0.447184
$66$	0	0
$67$	−9.76418	−1.19288	−0.596442	−	0.802656i	$-0.703420\pi$
$67$	−9.76418	−1.19288	−0.596442	+	0.802656i	$0.703420\pi$
$68$	0	0
$69$	7.69590	0.926477
$70$	0	0
$71$	−4.93177	−0.585293	−0.292647	−	0.956221i	$-0.594536\pi$
$71$	−4.93177	−0.585293	−0.292647	+	0.956221i	$0.594536\pi$
$72$	0	0
$73$	−9.25090	−1.08274	−0.541368	−	0.840786i	$-0.682093\pi$
$73$	−9.25090	−1.08274	−0.541368	+	0.840786i	$0.682093\pi$
$74$	0	0
$75$	−10.5998	−1.22396
$76$	0	0
$77$	−0.894467	−0.101934
$78$	0	0
$79$	4.83133	0.543567	0.271783	−	0.962358i	$-0.412387\pi$
$79$	4.83133	0.543567	0.271783	+	0.962358i	$0.412387\pi$
$80$	0	0
$81$	−10.6000	−1.17777
$82$	0	0
$83$	7.98855	0.876857	0.438429	−	0.898766i	$-0.355535\pi$
$83$	7.98855	0.876857	0.438429	+	0.898766i	$0.355535\pi$
$84$	0	0
$85$	2.14702	0.232877
$86$	0	0
$87$	1.08194	0.115996
$88$	0	0
$89$	−14.4971	−1.53669	−0.768347	−	0.640034i	$-0.778920\pi$
$89$	−14.4971	−1.53669	−0.768347	+	0.640034i	$0.778920\pi$
$90$	0	0
$91$	−1.52689	−0.160062
$92$	0	0
$93$	−8.81484	−0.914057
$94$	0	0
$95$	−0.631218	−0.0647616
$96$	0	0
$97$	12.1173	1.23032	0.615161	−	0.788402i	$-0.289091\pi$
$97$	12.1173	1.23032	0.615161	+	0.788402i	$0.289091\pi$
$98$	0	0
$99$	7.71659	0.775547
$100$	0	0
$101$	9.73898	0.969065	0.484533	−	0.874773i	$-0.338990\pi$
$101$	9.73898	0.969065	0.484533	+	0.874773i	$0.338990\pi$
$102$	0	0
$103$	4.91090	0.483885	0.241942	−	0.970291i	$-0.422215\pi$
$103$	4.91090	0.483885	0.241942	+	0.970291i	$0.422215\pi$
$104$	0	0
$105$	−0.388704	−0.0379336
$106$	0	0
$107$	0.734140	0.0709720	0.0354860	−	0.999370i	$-0.488702\pi$
$107$	0.734140	0.0709720	0.0354860	+	0.999370i	$0.488702\pi$
$108$	0	0
$109$	6.90153	0.661047	0.330523	−	0.943798i	$-0.392775\pi$
$109$	6.90153	0.661047	0.330523	+	0.943798i	$0.392775\pi$
$110$	0	0
$111$	−14.6434	−1.38989
$112$	0	0
$113$	−4.37583	−0.411643	−0.205822	−	0.978590i	$-0.565987\pi$
$113$	−4.37583	−0.411643	−0.205822	+	0.978590i	$0.565987\pi$
$114$	0	0
$115$	2.10885	0.196651
$116$	0	0
$117$	13.1726	1.21780
$118$	0	0
$119$	−0.909288	−0.0833543
$120$	0	0
$121$	0.195332	0.0177574
$122$	0	0
$123$	25.6196	2.31004
$124$	0	0
$125$	−6.06068	−0.542084
$126$	0	0
$127$	−3.66755	−0.325443	−0.162721	−	0.986672i	$-0.552027\pi$
$127$	−3.66755	−0.325443	−0.162721	+	0.986672i	$0.552027\pi$
$128$	0	0
$129$	15.2830	1.34559
$130$	0	0
$131$	11.0092	0.961880	0.480940	−	0.876753i	$-0.340295\pi$
$131$	11.0092	0.961880	0.480940	+	0.876753i	$0.340295\pi$
$132$	0	0
$133$	0.267329	0.0231803
$134$	0	0
$135$	−1.00873	−0.0868175
$136$	0	0
$137$	4.16481	0.355824	0.177912	−	0.984046i	$-0.443066\pi$
$137$	4.16481	0.355824	0.177912	+	0.984046i	$0.443066\pi$
$138$	0	0
$139$	11.1675	0.947216	0.473608	−	0.880736i	$-0.342951\pi$
$139$	11.1675	0.947216	0.473608	+	0.880736i	$0.342951\pi$
$140$	0	0
$141$	8.52579	0.718001
$142$	0	0
$143$	19.1109	1.59814
$144$	0	0
$145$	0.296475	0.0246209
$146$	0	0
$147$	−15.9601	−1.31637
$148$	0	0
$149$	5.36660	0.439649	0.219825	−	0.975539i	$-0.429451\pi$
$149$	5.36660	0.439649	0.219825	+	0.975539i	$0.429451\pi$
$150$	0	0
$151$	0.834281	0.0678928	0.0339464	−	0.999424i	$-0.489192\pi$
$151$	0.834281	0.0678928	0.0339464	+	0.999424i	$0.489192\pi$
$152$	0	0
$153$	7.84446	0.634187
$154$	0	0
$155$	−2.41546	−0.194014
$156$	0	0
$157$	4.08362	0.325908	0.162954	−	0.986634i	$-0.447898\pi$
$157$	4.08362	0.325908	0.162954	+	0.986634i	$0.447898\pi$
$158$	0	0
$159$	−2.30353	−0.182682
$160$	0	0
$161$	−0.893123	−0.0703879
$162$	0	0
$163$	−1.54241	−0.120811	−0.0604056	−	0.998174i	$-0.519239\pi$
$163$	−1.54241	−0.120811	−0.0604056	+	0.998174i	$0.519239\pi$
$164$	0	0
$165$	4.86510	0.378748
$166$	0	0
$167$	3.31614	0.256611	0.128305	−	0.991735i	$-0.459046\pi$
$167$	3.31614	0.256611	0.128305	+	0.991735i	$0.459046\pi$
$168$	0	0
$169$	19.6232	1.50947
$170$	0	0
$171$	−2.30625	−0.176364
$172$	0	0
$173$	11.1013	0.844014	0.422007	−	0.906593i	$-0.361326\pi$
$173$	11.1013	0.844014	0.422007	+	0.906593i	$0.361326\pi$
$174$	0	0
$175$	1.23013	0.0929891
$176$	0	0
$177$	1.88536	0.141713
$178$	0	0
$179$	−7.71730	−0.576818	−0.288409	−	0.957507i	$-0.593126\pi$
$179$	−7.71730	−0.576818	−0.288409	+	0.957507i	$0.593126\pi$
$180$	0	0
$181$	−15.4164	−1.14589	−0.572945	−	0.819594i	$-0.694199\pi$
$181$	−15.4164	−1.14589	−0.572945	+	0.819594i	$0.694199\pi$
$182$	0	0
$183$	2.92636	0.216323
$184$	0	0
$185$	−4.01261	−0.295013
$186$	0	0
$187$	11.3808	0.832250
$188$	0	0
$189$	0.427209	0.0310749
$190$	0	0
$191$	−12.1359	−0.878125	−0.439062	−	0.898457i	$-0.644689\pi$
$191$	−12.1359	−0.878125	−0.439062	+	0.898457i	$0.644689\pi$
$192$	0	0
$193$	13.9728	1.00579	0.502893	−	0.864349i	$-0.332269\pi$
$193$	13.9728	1.00579	0.502893	+	0.864349i	$0.332269\pi$
$194$	0	0
$195$	8.30494	0.594729
$196$	0	0
$197$	−10.6061	−0.755656	−0.377828	−	0.925876i	$-0.623329\pi$
$197$	−10.6061	−0.755656	−0.377828	+	0.925876i	$0.623329\pi$
$198$	0	0
$199$	8.99348	0.637531	0.318765	−	0.947834i	$-0.396732\pi$
$199$	8.99348	0.637531	0.318765	+	0.947834i	$0.396732\pi$
$200$	0	0
$201$	−22.4921	−1.58647
$202$	0	0
$203$	−0.125561	−0.00881264
$204$	0	0
$205$	7.02033	0.490321
$206$	0	0
$207$	7.70500	0.535534
$208$	0	0
$209$	−3.34594	−0.231444
$210$	0	0
$211$	−12.9058	−0.888472	−0.444236	−	0.895910i	$-0.646525\pi$
$211$	−12.9058	−0.888472	−0.444236	+	0.895910i	$0.646525\pi$
$212$	0	0
$213$	−11.3605	−0.778407
$214$	0	0
$215$	4.18787	0.285610
$216$	0	0
$217$	1.02298	0.0694443
$218$	0	0
$219$	−21.3097	−1.43998
$220$	0	0
$221$	19.4276	1.30684
$222$	0	0
$223$	13.0823	0.876053	0.438026	−	0.898962i	$-0.355678\pi$
$223$	13.0823	0.876053	0.438026	+	0.898962i	$0.355678\pi$
$224$	0	0
$225$	−10.6124	−0.707492
$226$	0	0
$227$	−18.3158	−1.21566	−0.607830	−	0.794067i	$-0.707960\pi$
$227$	−18.3158	−1.21566	−0.607830	+	0.794067i	$0.707960\pi$
$228$	0	0
$229$	14.3785	0.950158	0.475079	−	0.879943i	$-0.342420\pi$
$229$	14.3785	0.950158	0.475079	+	0.879943i	$0.342420\pi$
$230$	0	0
$231$	−2.06043	−0.135566
$232$	0	0
$233$	−13.4929	−0.883950	−0.441975	−	0.897027i	$-0.645722\pi$
$233$	−13.4929	−0.883950	−0.441975	+	0.897027i	$0.645722\pi$
$234$	0	0
$235$	2.33626	0.152400
$236$	0	0
$237$	11.1291	0.722913
$238$	0	0
$239$	−4.19997	−0.271674	−0.135837	−	0.990731i	$-0.543372\pi$
$239$	−4.19997	−0.271674	−0.135837	+	0.990731i	$0.543372\pi$
$240$	0	0
$241$	2.09976	0.135257	0.0676286	−	0.997711i	$-0.478457\pi$
$241$	2.09976	0.135257	0.0676286	+	0.997711i	$0.478457\pi$
$242$	0	0
$243$	−19.6231	−1.25882
$244$	0	0
$245$	−4.37342	−0.279407
$246$	0	0
$247$	−5.71167	−0.363425
$248$	0	0
$249$	18.4019	1.16617
$250$	0	0
$251$	−11.4942	−0.725507	−0.362754	−	0.931885i	$-0.618163\pi$
$251$	−11.4942	−0.725507	−0.362754	+	0.931885i	$0.618163\pi$
$252$	0	0
$253$	11.1785	0.702787
$254$	0	0
$255$	4.94572	0.309713
$256$	0	0
$257$	−19.0533	−1.18851	−0.594256	−	0.804276i	$-0.702553\pi$
$257$	−19.0533	−1.18851	−0.594256	+	0.804276i	$0.702553\pi$
$258$	0	0
$259$	1.69939	0.105595
$260$	0	0
$261$	1.08322	0.0670495
$262$	0	0
$263$	0.793033	0.0489005	0.0244502	−	0.999701i	$-0.492216\pi$
$263$	0.793033	0.0489005	0.0244502	+	0.999701i	$0.492216\pi$
$264$	0	0
$265$	−0.631218	−0.0387754
$266$	0	0
$267$	−33.3946	−2.04372
$268$	0	0
$269$	2.63786	0.160833	0.0804166	−	0.996761i	$-0.474375\pi$
$269$	2.63786	0.160833	0.0804166	+	0.996761i	$0.474375\pi$
$270$	0	0
$271$	6.25455	0.379937	0.189968	−	0.981790i	$-0.439161\pi$
$271$	6.25455	0.379937	0.189968	+	0.981790i	$0.439161\pi$
$272$	0	0
$273$	−3.51725	−0.212873
$274$	0	0
$275$	−15.3966	−0.928448
$276$	0	0
$277$	−3.37484	−0.202774	−0.101387	−	0.994847i	$-0.532328\pi$
$277$	−3.37484	−0.202774	−0.101387	+	0.994847i	$0.532328\pi$
$278$	0	0
$279$	−8.82526	−0.528355
$280$	0	0
$281$	7.82052	0.466533	0.233267	−	0.972413i	$-0.425059\pi$
$281$	7.82052	0.466533	0.233267	+	0.972413i	$0.425059\pi$
$282$	0	0
$283$	0.716104	0.0425680	0.0212840	−	0.999773i	$-0.493225\pi$
$283$	0.716104	0.0425680	0.0212840	+	0.999773i	$0.493225\pi$
$284$	0	0
$285$	−1.45403	−0.0861293
$286$	0	0
$287$	−2.97320	−0.175502
$288$	0	0
$289$	−5.43058	−0.319446
$290$	0	0
$291$	27.9125	1.63626
$292$	0	0
$293$	−6.68499	−0.390541	−0.195271	−	0.980749i	$-0.562558\pi$
$293$	−6.68499	−0.390541	−0.195271	+	0.980749i	$0.562558\pi$
$294$	0	0
$295$	0.516631	0.0300794
$296$	0	0
$297$	−5.34704	−0.310267
$298$	0	0
$299$	19.0822	1.10355
$300$	0	0
$301$	−1.77362	−0.102229
$302$	0	0
$303$	22.4340	1.28880
$304$	0	0
$305$	0.801888	0.0459160
$306$	0	0
$307$	−5.78171	−0.329979	−0.164990	−	0.986295i	$-0.552759\pi$
$307$	−5.78171	−0.329979	−0.164990	+	0.986295i	$0.552759\pi$
$308$	0	0
$309$	11.3124	0.643540
$310$	0	0
$311$	29.1867	1.65502	0.827512	−	0.561448i	$-0.189755\pi$
$311$	29.1867	1.65502	0.827512	+	0.561448i	$0.189755\pi$
$312$	0	0
$313$	−19.0626	−1.07748	−0.538740	−	0.842472i	$-0.681100\pi$
$313$	−19.0626	−1.07748	−0.538740	+	0.842472i	$0.681100\pi$
$314$	0	0
$315$	−0.389164	−0.0219269
$316$	0	0
$317$	−14.2926	−0.802753	−0.401376	−	0.915913i	$-0.631468\pi$
$317$	−14.2926	−0.802753	−0.401376	+	0.915913i	$0.631468\pi$
$318$	0	0
$319$	1.57155	0.0879897
$320$	0	0
$321$	1.69111	0.0943888
$322$	0	0
$323$	−3.40139	−0.189258
$324$	0	0
$325$	−26.2826	−1.45790
$326$	0	0
$327$	15.8979	0.879155
$328$	0	0
$329$	−0.989434	−0.0545492
$330$	0	0
$331$	5.74257	0.315640	0.157820	−	0.987468i	$-0.449553\pi$
$331$	5.74257	0.315640	0.157820	+	0.987468i	$0.449553\pi$
$332$	0	0
$333$	−14.6607	−0.803402
$334$	0	0
$335$	−6.16333	−0.336739
$336$	0	0
$337$	7.06769	0.385001	0.192501	−	0.981297i	$-0.438340\pi$
$337$	7.06769	0.385001	0.192501	+	0.981297i	$0.438340\pi$
$338$	0	0
$339$	−10.0799	−0.547463
$340$	0	0
$341$	−12.8038	−0.693365
$342$	0	0
$343$	3.72350	0.201050
$344$	0	0
$345$	4.85779	0.261535
$346$	0	0
$347$	−31.8342	−1.70895	−0.854474	−	0.519494i	$-0.826120\pi$
$347$	−31.8342	−1.70895	−0.854474	+	0.519494i	$0.826120\pi$
$348$	0	0
$349$	−1.53584	−0.0822114	−0.0411057	−	0.999155i	$-0.513088\pi$
$349$	−1.53584	−0.0822114	−0.0411057	+	0.999155i	$0.513088\pi$
$350$	0	0
$351$	−9.12762	−0.487197
$352$	0	0
$353$	22.7507	1.21090	0.605448	−	0.795885i	$-0.292994\pi$
$353$	22.7507	1.21090	0.605448	+	0.795885i	$0.292994\pi$
$354$	0	0
$355$	−3.11302	−0.165222
$356$	0	0
$357$	−2.09457	−0.110857
$358$	0	0
$359$	21.0677	1.11191	0.555956	−	0.831212i	$-0.312352\pi$
$359$	21.0677	1.11191	0.555956	+	0.831212i	$0.312352\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.449953	0.0236164
$364$	0	0
$365$	−5.83933	−0.305645
$366$	0	0
$367$	−26.0247	−1.35848	−0.679239	−	0.733917i	$-0.737690\pi$
$367$	−26.0247	−1.35848	−0.679239	+	0.733917i	$0.737690\pi$
$368$	0	0
$369$	25.6499	1.33528
$370$	0	0
$371$	0.267329	0.0138790
$372$	0	0
$373$	−22.5967	−1.17001	−0.585006	−	0.811029i	$-0.698908\pi$
$373$	−22.5967	−1.17001	−0.585006	+	0.811029i	$0.698908\pi$
$374$	0	0
$375$	−13.9610	−0.720941
$376$	0	0
$377$	2.68270	0.138166
$378$	0	0
$379$	−17.2383	−0.885469	−0.442735	−	0.896653i	$-0.645992\pi$
$379$	−17.2383	−0.885469	−0.442735	+	0.896653i	$0.645992\pi$
$380$	0	0
$381$	−8.44832	−0.432821
$382$	0	0
$383$	−6.67634	−0.341145	−0.170573	−	0.985345i	$-0.554562\pi$
$383$	−6.67634	−0.341145	−0.170573	+	0.985345i	$0.554562\pi$
$384$	0	0
$385$	−0.564604	−0.0287749
$386$	0	0
$387$	15.3010	0.777795
$388$	0	0
$389$	−25.4787	−1.29182	−0.645910	−	0.763413i	$-0.723522\pi$
$389$	−25.4787	−1.29182	−0.645910	+	0.763413i	$0.723522\pi$
$390$	0	0
$391$	11.3637	0.574689
$392$	0	0
$393$	25.3601	1.27925
$394$	0	0
$395$	3.04962	0.153443
$396$	0	0
$397$	−20.7359	−1.04071	−0.520353	−	0.853951i	$-0.674200\pi$
$397$	−20.7359	−1.04071	−0.520353	+	0.853951i	$0.674200\pi$
$398$	0	0
$399$	0.615800	0.0308286
$400$	0	0
$401$	−0.557390	−0.0278347	−0.0139174	−	0.999903i	$-0.504430\pi$
$401$	−0.557390	−0.0278347	−0.0139174	+	0.999903i	$0.504430\pi$
$402$	0	0
$403$	−21.8566	−1.08876
$404$	0	0
$405$	−6.69088	−0.332473
$406$	0	0
$407$	−21.2699	−1.05431
$408$	0	0
$409$	−3.42613	−0.169411	−0.0847057	−	0.996406i	$-0.526995\pi$
$409$	−3.42613	−0.169411	−0.0847057	+	0.996406i	$0.526995\pi$
$410$	0	0
$411$	9.59377	0.473226
$412$	0	0
$413$	−0.218800	−0.0107664
$414$	0	0
$415$	5.04252	0.247527
$416$	0	0
$417$	25.7247	1.25974
$418$	0	0
$419$	12.9480	0.632552	0.316276	−	0.948667i	$-0.397568\pi$
$419$	12.9480	0.632552	0.316276	+	0.948667i	$0.397568\pi$
$420$	0	0
$421$	−0.0178434	−0.000869633 0	−0.000434816	−	1.00000i	$-0.500138\pi$
$421$	−0.0178434	−0.000869633 0	−0.000434816		1.00000i	$0.500138\pi$
$422$	0	0
$423$	8.53587	0.415028
$424$	0	0
$425$	−15.6517	−0.759219
$426$	0	0
$427$	−0.339610	−0.0164349
$428$	0	0
$429$	44.0226	2.12543
$430$	0	0
$431$	−12.0145	−0.578717	−0.289358	−	0.957221i	$-0.593442\pi$
$431$	−12.0145	−0.578717	−0.289358	+	0.957221i	$0.593442\pi$
$432$	0	0
$433$	12.9964	0.624565	0.312282	−	0.949989i	$-0.398906\pi$
$433$	12.9964	0.624565	0.312282	+	0.949989i	$0.398906\pi$
$434$	0	0
$435$	0.682939	0.0327444
$436$	0	0
$437$	−3.34091	−0.159818
$438$	0	0
$439$	15.8147	0.754793	0.377397	−	0.926052i	$-0.376819\pi$
$439$	15.8147	0.754793	0.377397	+	0.926052i	$0.376819\pi$
$440$	0	0
$441$	−15.9790	−0.760903
$442$	0	0
$443$	−5.81123	−0.276100	−0.138050	−	0.990425i	$-0.544083\pi$
$443$	−5.81123	−0.276100	−0.138050	+	0.990425i	$0.544083\pi$
$444$	0	0
$445$	−9.15085	−0.433792
$446$	0	0
$447$	12.3621	0.584709
$448$	0	0
$449$	−14.3300	−0.676277	−0.338138	−	0.941096i	$-0.609797\pi$
$449$	−14.3300	−0.676277	−0.338138	+	0.941096i	$0.609797\pi$
$450$	0	0
$451$	37.2131	1.75230
$452$	0	0
$453$	1.92179	0.0902937
$454$	0	0
$455$	−0.963803	−0.0451838
$456$	0	0
$457$	19.4826	0.911356	0.455678	−	0.890145i	$-0.349397\pi$
$457$	19.4826	0.911356	0.455678	+	0.890145i	$0.349397\pi$
$458$	0	0
$459$	−5.43564	−0.253714
$460$	0	0
$461$	−21.6503	−1.00835	−0.504177	−	0.863600i	$-0.668204\pi$
$461$	−21.6503	−1.00835	−0.504177	+	0.863600i	$0.668204\pi$
$462$	0	0
$463$	2.06332	0.0958907	0.0479454	−	0.998850i	$-0.484733\pi$
$463$	2.06332	0.0958907	0.0479454	+	0.998850i	$0.484733\pi$
$464$	0	0
$465$	−5.56409	−0.258028
$466$	0	0
$467$	−6.83163	−0.316130	−0.158065	−	0.987429i	$-0.550526\pi$
$467$	−6.83163	−0.316130	−0.158065	+	0.987429i	$0.550526\pi$
$468$	0	0
$469$	2.61025	0.120530
$470$	0	0
$471$	9.40673	0.433439
$472$	0	0
$473$	22.1989	1.02071
$474$	0	0
$475$	4.60156	0.211134
$476$	0	0
$477$	−2.30625	−0.105596
$478$	0	0
$479$	−7.23252	−0.330462	−0.165231	−	0.986255i	$-0.552837\pi$
$479$	−7.23252	−0.330462	−0.165231	+	0.986255i	$0.552837\pi$
$480$	0	0
$481$	−36.3087	−1.65553
$482$	0	0
$483$	−2.05734	−0.0936120
$484$	0	0
$485$	7.64864	0.347307
$486$	0	0
$487$	8.97017	0.406477	0.203239	−	0.979129i	$-0.434853\pi$
$487$	8.97017	0.406477	0.203239	+	0.979129i	$0.434853\pi$
$488$	0	0
$489$	−3.55300	−0.160672
$490$	0	0
$491$	−10.0184	−0.452126	−0.226063	−	0.974113i	$-0.572586\pi$
$491$	−10.0184	−0.452126	−0.226063	+	0.974113i	$0.572586\pi$
$492$	0	0
$493$	1.59759	0.0719517
$494$	0	0
$495$	4.87085	0.218929
$496$	0	0
$497$	1.31840	0.0591385
$498$	0	0
$499$	−14.9432	−0.668949	−0.334474	−	0.942405i	$-0.608559\pi$
$499$	−14.9432	−0.668949	−0.334474	+	0.942405i	$0.608559\pi$
$500$	0	0
$501$	7.63884	0.341278
$502$	0	0
$503$	21.4761	0.957571	0.478786	−	0.877932i	$-0.341077\pi$
$503$	21.4761	0.957571	0.478786	+	0.877932i	$0.341077\pi$
$504$	0	0
$505$	6.14742	0.273557
$506$	0	0
$507$	45.2026	2.00752
$508$	0	0
$509$	3.36972	0.149360	0.0746801	−	0.997208i	$-0.476206\pi$
$509$	3.36972	0.149360	0.0746801	+	0.997208i	$0.476206\pi$
$510$	0	0
$511$	2.47303	0.109400
$512$	0	0
$513$	1.59807	0.0705563
$514$	0	0
$515$	3.09985	0.136596
$516$	0	0
$517$	12.3840	0.544646
$518$	0	0
$519$	25.5721	1.12249
$520$	0	0
$521$	−2.41306	−0.105718	−0.0528591	−	0.998602i	$-0.516833\pi$
$521$	−2.41306	−0.105718	−0.0528591	+	0.998602i	$0.516833\pi$
$522$	0	0
$523$	−37.1015	−1.62234	−0.811169	−	0.584812i	$-0.801168\pi$
$523$	−37.1015	−1.62234	−0.811169	+	0.584812i	$0.801168\pi$
$524$	0	0
$525$	2.83364	0.123670
$526$	0	0
$527$	−13.0160	−0.566984
$528$	0	0
$529$	−11.8383	−0.514708
$530$	0	0
$531$	1.88759	0.0819145
$532$	0	0
$533$	63.5244	2.75155
$534$	0	0
$535$	0.463403	0.0200346
$536$	0	0
$537$	−17.7770	−0.767136
$538$	0	0
$539$	−23.1825	−0.998540
$540$	0	0
$541$	5.58275	0.240021	0.120011	−	0.992773i	$-0.461707\pi$
$541$	5.58275	0.240021	0.120011	+	0.992773i	$0.461707\pi$
$542$	0	0
$543$	−35.5121	−1.52397
$544$	0	0
$545$	4.35637	0.186606
$546$	0	0
$547$	39.6554	1.69554	0.847771	−	0.530362i	$-0.177944\pi$
$547$	39.6554	1.69554	0.847771	+	0.530362i	$0.177944\pi$
$548$	0	0
$549$	2.92982	0.125042
$550$	0	0
$551$	−0.469687	−0.0200093
$552$	0	0
$553$	−1.29155	−0.0549224
$554$	0	0
$555$	−9.24318	−0.392351
$556$	0	0
$557$	33.0050	1.39846	0.699232	−	0.714894i	$-0.253525\pi$
$557$	33.0050	1.39846	0.699232	+	0.714894i	$0.253525\pi$
$558$	0	0
$559$	37.8945	1.60277
$560$	0	0
$561$	26.2161	1.10685
$562$	0	0
$563$	−30.4970	−1.28529	−0.642647	−	0.766162i	$-0.722164\pi$
$563$	−30.4970	−1.28529	−0.642647	+	0.766162i	$0.722164\pi$
$564$	0	0
$565$	−2.76210	−0.116203
$566$	0	0
$567$	2.83367	0.119003
$568$	0	0
$569$	−20.5370	−0.860957	−0.430479	−	0.902601i	$-0.641655\pi$
$569$	−20.5370	−0.860957	−0.430479	+	0.902601i	$0.641655\pi$
$570$	0	0
$571$	−20.3183	−0.850293	−0.425146	−	0.905125i	$-0.639777\pi$
$571$	−20.3183	−0.850293	−0.425146	+	0.905125i	$0.639777\pi$
$572$	0	0
$573$	−27.9555	−1.16786
$574$	0	0
$575$	−15.3734	−0.641116
$576$	0	0
$577$	−27.9211	−1.16237	−0.581185	−	0.813771i	$-0.697411\pi$
$577$	−27.9211	−1.16237	−0.581185	+	0.813771i	$0.697411\pi$
$578$	0	0
$579$	32.1868	1.33764
$580$	0	0
$581$	−2.13557	−0.0885984
$582$	0	0
$583$	−3.34594	−0.138575
$584$	0	0
$585$	8.31476	0.343773
$586$	0	0
$587$	−32.4871	−1.34089	−0.670443	−	0.741961i	$-0.733896\pi$
$587$	−32.4871	−1.34089	−0.670443	+	0.741961i	$0.733896\pi$
$588$	0	0
$589$	3.82667	0.157675
$590$	0	0
$591$	−24.4316	−1.00498
$592$	0	0
$593$	22.1512	0.909639	0.454819	−	0.890584i	$-0.349704\pi$
$593$	22.1512	0.909639	0.454819	+	0.890584i	$0.349704\pi$
$594$	0	0
$595$	−0.573959	−0.0235300
$596$	0	0
$597$	20.7168	0.847880
$598$	0	0
$599$	−22.3297	−0.912366	−0.456183	−	0.889886i	$-0.650784\pi$
$599$	−22.3297	−0.912366	−0.456183	+	0.889886i	$0.650784\pi$
$600$	0	0
$601$	20.5017	0.836281	0.418140	−	0.908382i	$-0.362682\pi$
$601$	20.5017	0.836281	0.418140	+	0.908382i	$0.362682\pi$
$602$	0	0
$603$	−22.5187	−0.917031
$604$	0	0
$605$	0.123297	0.00501273
$606$	0	0
$607$	14.2890	0.579971	0.289985	−	0.957031i	$-0.406350\pi$
$607$	14.2890	0.579971	0.289985	+	0.957031i	$0.406350\pi$
$608$	0	0
$609$	−0.289233	−0.0117203
$610$	0	0
$611$	21.1399	0.855231
$612$	0	0
$613$	−24.4641	−0.988095	−0.494048	−	0.869435i	$-0.664483\pi$
$613$	−24.4641	−0.988095	−0.494048	+	0.869435i	$0.664483\pi$
$614$	0	0
$615$	16.1715	0.652099
$616$	0	0
$617$	−45.1294	−1.81684	−0.908421	−	0.418056i	$-0.862712\pi$
$617$	−45.1294	−1.81684	−0.908421	+	0.418056i	$0.862712\pi$
$618$	0	0
$619$	14.5558	0.585048	0.292524	−	0.956258i	$-0.405505\pi$
$619$	14.5558	0.585048	0.292524	+	0.956258i	$0.405505\pi$
$620$	0	0
$621$	−5.33900	−0.214247
$622$	0	0
$623$	3.87550	0.155269
$624$	0	0
$625$	19.1822	0.767288
$626$	0	0
$627$	−7.70748	−0.307807
$628$	0	0
$629$	−21.6224	−0.862141
$630$	0	0
$631$	9.56597	0.380815	0.190408	−	0.981705i	$-0.439019\pi$
$631$	9.56597	0.380815	0.190408	+	0.981705i	$0.439019\pi$
$632$	0	0
$633$	−29.7289	−1.18162
$634$	0	0
$635$	−2.31503	−0.0918690
$636$	0	0
$637$	−39.5735	−1.56796
$638$	0	0
$639$	−11.3739	−0.449945
$640$	0	0
$641$	−11.6152	−0.458773	−0.229387	−	0.973335i	$-0.573672\pi$
$641$	−11.6152	−0.458773	−0.229387	+	0.973335i	$0.573672\pi$
$642$	0	0
$643$	34.7981	1.37230	0.686151	−	0.727459i	$-0.259299\pi$
$643$	34.7981	1.37230	0.686151	+	0.727459i	$0.259299\pi$
$644$	0	0
$645$	9.64688	0.379846
$646$	0	0
$647$	8.85274	0.348037	0.174018	−	0.984742i	$-0.444325\pi$
$647$	8.85274	0.348037	0.174018	+	0.984742i	$0.444325\pi$
$648$	0	0
$649$	2.73854	0.107497
$650$	0	0
$651$	2.35646	0.0923570
$652$	0	0
$653$	1.82503	0.0714188	0.0357094	−	0.999362i	$-0.488631\pi$
$653$	1.82503	0.0714188	0.0357094	+	0.999362i	$0.488631\pi$
$654$	0	0
$655$	6.94922	0.271529
$656$	0	0
$657$	−21.3349	−0.832354
$658$	0	0
$659$	5.67165	0.220936	0.110468	−	0.993880i	$-0.464765\pi$
$659$	5.67165	0.220936	0.110468	+	0.993880i	$0.464765\pi$
$660$	0	0
$661$	−23.2423	−0.904021	−0.452010	−	0.892013i	$-0.649293\pi$
$661$	−23.2423	−0.904021	−0.452010	+	0.892013i	$0.649293\pi$
$662$	0	0
$663$	44.7521	1.73803
$664$	0	0
$665$	0.168743	0.00654357
$666$	0	0
$667$	1.56918	0.0607590
$668$	0	0
$669$	30.1354	1.16510
$670$	0	0
$671$	4.25062	0.164093
$672$	0	0
$673$	−44.9147	−1.73133	−0.865666	−	0.500621i	$-0.833105\pi$
$673$	−44.9147	−1.73133	−0.865666	+	0.500621i	$0.833105\pi$
$674$	0	0
$675$	7.35360	0.283040
$676$	0	0
$677$	33.6177	1.29203	0.646017	−	0.763323i	$-0.276434\pi$
$677$	33.6177	1.29203	0.646017	+	0.763323i	$0.276434\pi$
$678$	0	0
$679$	−3.23929	−0.124313
$680$	0	0
$681$	−42.1909	−1.61676
$682$	0	0
$683$	−8.32318	−0.318478	−0.159239	−	0.987240i	$-0.550904\pi$
$683$	−8.32318	−0.318478	−0.159239	+	0.987240i	$0.550904\pi$
$684$	0	0
$685$	2.62890	0.100445
$686$	0	0
$687$	33.1213	1.26366
$688$	0	0
$689$	−5.71167	−0.217597
$690$	0	0
$691$	28.1008	1.06900	0.534502	−	0.845167i	$-0.320499\pi$
$691$	28.1008	1.06900	0.534502	+	0.845167i	$0.320499\pi$
$692$	0	0
$693$	−2.06287	−0.0783619
$694$	0	0
$695$	7.04914	0.267389
$696$	0	0
$697$	37.8298	1.43290
$698$	0	0
$699$	−31.0814	−1.17560
$700$	0	0
$701$	31.3722	1.18491	0.592456	−	0.805603i	$-0.298158\pi$
$701$	31.3722	1.18491	0.592456	+	0.805603i	$0.298158\pi$
$702$	0	0
$703$	6.35693	0.239756
$704$	0	0
$705$	5.38164	0.202684
$706$	0	0
$707$	−2.60351	−0.0979151
$708$	0	0
$709$	−47.6306	−1.78880	−0.894402	−	0.447265i	$-0.852398\pi$
$709$	−47.6306	−1.78880	−0.894402	+	0.447265i	$0.852398\pi$
$710$	0	0
$711$	11.1423	0.417868
$712$	0	0
$713$	−12.7846	−0.478786
$714$	0	0
$715$	12.0632	0.451137
$716$	0	0
$717$	−9.67477	−0.361311
$718$	0	0
$719$	37.4975	1.39842	0.699211	−	0.714916i	$-0.253535\pi$
$719$	37.4975	1.39842	0.699211	+	0.714916i	$0.253535\pi$
$720$	0	0
$721$	−1.31282	−0.0488921
$722$	0	0
$723$	4.83685	0.179884
$724$	0	0
$725$	−2.16129	−0.0802685
$726$	0	0
$727$	−2.82137	−0.104639	−0.0523193	−	0.998630i	$-0.516661\pi$
$727$	−2.82137	−0.104639	−0.0523193	+	0.998630i	$0.516661\pi$
$728$	0	0
$729$	−13.4026	−0.496393
$730$	0	0
$731$	22.5668	0.834662
$732$	0	0
$733$	−37.3748	−1.38047	−0.690236	−	0.723585i	$-0.742493\pi$
$733$	−37.3748	−1.38047	−0.690236	+	0.723585i	$0.742493\pi$
$734$	0	0
$735$	−10.0743	−0.371596
$736$	0	0
$737$	−32.6704	−1.20343
$738$	0	0
$739$	−9.35332	−0.344068	−0.172034	−	0.985091i	$-0.555034\pi$
$739$	−9.35332	−0.344068	−0.172034	+	0.985091i	$0.555034\pi$
$740$	0	0
$741$	−13.1570	−0.483335
$742$	0	0
$743$	29.0252	1.06483	0.532415	−	0.846483i	$-0.321284\pi$
$743$	29.0252	1.06483	0.532415	+	0.846483i	$0.321284\pi$
$744$	0	0
$745$	3.38750	0.124108
$746$	0	0
$747$	18.4236	0.674085
$748$	0	0
$749$	−0.196257	−0.00717107
$750$	0	0
$751$	16.7860	0.612529	0.306264	−	0.951946i	$-0.400921\pi$
$751$	16.7860	0.612529	0.306264	+	0.951946i	$0.400921\pi$
$752$	0	0
$753$	−26.4772	−0.964884
$754$	0	0
$755$	0.526613	0.0191654
$756$	0	0
$757$	−8.86976	−0.322377	−0.161188	−	0.986924i	$-0.551533\pi$
$757$	−8.86976	−0.322377	−0.161188	+	0.986924i	$0.551533\pi$
$758$	0	0
$759$	25.7500	0.934667
$760$	0	0
$761$	−17.5852	−0.637461	−0.318731	−	0.947845i	$-0.603257\pi$
$761$	−17.5852	−0.637461	−0.318731	+	0.947845i	$0.603257\pi$
$762$	0	0
$763$	−1.84498	−0.0667927
$764$	0	0
$765$	4.95156	0.179024
$766$	0	0
$767$	4.67481	0.168798
$768$	0	0
$769$	−6.77466	−0.244301	−0.122150	−	0.992512i	$-0.538979\pi$
$769$	−6.77466	−0.244301	−0.122150	+	0.992512i	$0.538979\pi$
$770$	0	0
$771$	−43.8898	−1.58065
$772$	0	0
$773$	10.9553	0.394034	0.197017	−	0.980400i	$-0.436875\pi$
$773$	10.9553	0.394034	0.197017	+	0.980400i	$0.436875\pi$
$774$	0	0
$775$	17.6086	0.632521
$776$	0	0
$777$	3.91460	0.140436
$778$	0	0
$779$	−11.1219	−0.398482
$780$	0	0
$781$	−16.5014	−0.590467
$782$	0	0
$783$	−0.750591	−0.0268239
$784$	0	0
$785$	2.57765	0.0920003
$786$	0	0
$787$	−14.6957	−0.523844	−0.261922	−	0.965089i	$-0.584356\pi$
$787$	−14.6957	−0.523844	−0.261922	+	0.965089i	$0.584356\pi$
$788$	0	0
$789$	1.82678	0.0650349
$790$	0	0
$791$	1.16979	0.0415928
$792$	0	0
$793$	7.25600	0.257668
$794$	0	0
$795$	−1.45403	−0.0515691
$796$	0	0
$797$	−35.6843	−1.26400	−0.632001	−	0.774968i	$-0.717766\pi$
$797$	−35.6843	−1.26400	−0.632001	+	0.774968i	$0.717766\pi$
$798$	0	0
$799$	12.5892	0.445372
$800$	0	0
$801$	−33.4341	−1.18133
$802$	0	0
$803$	−30.9530	−1.09231
$804$	0	0
$805$	−0.563755	−0.0198698
$806$	0	0
$807$	6.07639	0.213899
$808$	0	0
$809$	−35.7281	−1.25613	−0.628067	−	0.778159i	$-0.716154\pi$
$809$	−35.7281	−1.25613	−0.628067	+	0.778159i	$0.716154\pi$
$810$	0	0
$811$	23.6862	0.831734	0.415867	−	0.909425i	$-0.363478\pi$
$811$	23.6862	0.831734	0.415867	+	0.909425i	$0.363478\pi$
$812$	0	0
$813$	14.4075	0.505295
$814$	0	0
$815$	−0.973600	−0.0341037
$816$	0	0
$817$	−6.63458	−0.232115
$818$	0	0
$819$	−3.52140	−0.123048
$820$	0	0
$821$	37.4766	1.30794	0.653971	−	0.756520i	$-0.273102\pi$
$821$	37.4766	1.30794	0.653971	+	0.756520i	$0.273102\pi$
$822$	0	0
$823$	−0.647096	−0.0225563	−0.0112782	−	0.999936i	$-0.503590\pi$
$823$	−0.647096	−0.0225563	−0.0112782	+	0.999936i	$0.503590\pi$
$824$	0	0
$825$	−35.4665	−1.23478
$826$	0	0
$827$	−12.2302	−0.425286	−0.212643	−	0.977130i	$-0.568207\pi$
$827$	−12.2302	−0.425286	−0.212643	+	0.977130i	$0.568207\pi$
$828$	0	0
$829$	4.27222	0.148380	0.0741901	−	0.997244i	$-0.476363\pi$
$829$	4.27222	0.148380	0.0741901	+	0.997244i	$0.476363\pi$
$830$	0	0
$831$	−7.77404	−0.269678
$832$	0	0
$833$	−23.5666	−0.816535
$834$	0	0
$835$	2.09321	0.0724385
$836$	0	0
$837$	6.11526	0.211374
$838$	0	0
$839$	27.2823	0.941890	0.470945	−	0.882163i	$-0.343913\pi$
$839$	27.2823	0.941890	0.470945	+	0.882163i	$0.343913\pi$
$840$	0	0
$841$	−28.7794	−0.992393
$842$	0	0
$843$	18.0148	0.620463
$844$	0	0
$845$	12.3865	0.426109
$846$	0	0
$847$	−0.0522178	−0.00179422
$848$	0	0
$849$	1.64957	0.0566130
$850$	0	0
$851$	−21.2380	−0.728028
$852$	0	0
$853$	24.2737	0.831116	0.415558	−	0.909567i	$-0.363586\pi$
$853$	24.2737	0.831116	0.415558	+	0.909567i	$0.363586\pi$
$854$	0	0
$855$	−1.45575	−0.0497856
$856$	0	0
$857$	−22.0284	−0.752475	−0.376237	−	0.926523i	$-0.622782\pi$
$857$	−22.0284	−0.752475	−0.376237	+	0.926523i	$0.622782\pi$
$858$	0	0
$859$	−8.95736	−0.305621	−0.152811	−	0.988256i	$-0.548832\pi$
$859$	−8.95736	−0.305621	−0.152811	+	0.988256i	$0.548832\pi$
$860$	0	0
$861$	−6.84885	−0.233408
$862$	0	0
$863$	−34.2724	−1.16664	−0.583322	−	0.812241i	$-0.698248\pi$
$863$	−34.2724	−1.16664	−0.583322	+	0.812241i	$0.698248\pi$
$864$	0	0
$865$	7.00732	0.238256
$866$	0	0
$867$	−12.5095	−0.424845
$868$	0	0
$869$	16.1653	0.548372
$870$	0	0
$871$	−55.7698	−1.88969
$872$	0	0
$873$	27.9455	0.945811
$874$	0	0
$875$	1.62019	0.0547726
$876$	0	0
$877$	−17.2731	−0.583271	−0.291636	−	0.956529i	$-0.594199\pi$
$877$	−17.2731	−0.583271	−0.291636	+	0.956529i	$0.594199\pi$
$878$	0	0
$879$	−15.3991	−0.519398
$880$	0	0
$881$	21.1584	0.712845	0.356422	−	0.934325i	$-0.383996\pi$
$881$	21.1584	0.712845	0.356422	+	0.934325i	$0.383996\pi$
$882$	0	0
$883$	32.6380	1.09836	0.549178	−	0.835706i	$-0.314941\pi$
$883$	32.6380	1.09836	0.549178	+	0.835706i	$0.314941\pi$
$884$	0	0
$885$	1.19008	0.0400039
$886$	0	0
$887$	17.7542	0.596127	0.298063	−	0.954546i	$-0.403659\pi$
$887$	17.7542	0.596127	0.298063	+	0.954546i	$0.403659\pi$
$888$	0	0
$889$	0.980443	0.0328830
$890$	0	0
$891$	−35.4668	−1.18818
$892$	0	0
$893$	−3.70119	−0.123855
$894$	0	0
$895$	−4.87130	−0.162830
$896$	0	0
$897$	43.9564	1.46766
$898$	0	0
$899$	−1.79733	−0.0599445
$900$	0	0
$901$	−3.40139	−0.113317
$902$	0	0
$903$	−4.08558	−0.135959
$904$	0	0
$905$	−9.73110	−0.323473
$906$	0	0
$907$	29.8497	0.991144	0.495572	−	0.868567i	$-0.334958\pi$
$907$	29.8497	0.991144	0.495572	+	0.868567i	$0.334958\pi$
$908$	0	0
$909$	22.4606	0.744970
$910$	0	0
$911$	−22.0073	−0.729134	−0.364567	−	0.931177i	$-0.618783\pi$
$911$	−22.0073	−0.729134	−0.364567	+	0.931177i	$0.618783\pi$
$912$	0	0
$913$	26.7292	0.884608
$914$	0	0
$915$	1.84717	0.0610657
$916$	0	0
$917$	−2.94308	−0.0971891
$918$	0	0
$919$	26.0180	0.858255	0.429127	−	0.903244i	$-0.358821\pi$
$919$	26.0180	0.858255	0.429127	+	0.903244i	$0.358821\pi$
$920$	0	0
$921$	−13.3183	−0.438854
$922$	0	0
$923$	−28.1686	−0.927182
$924$	0	0
$925$	29.2518	0.961795
$926$	0	0
$927$	11.3258	0.371987
$928$	0	0
$929$	14.3857	0.471981	0.235990	−	0.971755i	$-0.424167\pi$
$929$	14.3857	0.471981	0.235990	+	0.971755i	$0.424167\pi$
$930$	0	0
$931$	6.92854	0.227074
$932$	0	0
$933$	67.2324	2.20109
$934$	0	0
$935$	7.18379	0.234935
$936$	0	0
$937$	−47.4554	−1.55030	−0.775150	−	0.631778i	$-0.782326\pi$
$937$	−47.4554	−1.55030	−0.775150	+	0.631778i	$0.782326\pi$
$938$	0	0
$939$	−43.9112	−1.43299
$940$	0	0
$941$	13.2728	0.432682	0.216341	−	0.976318i	$-0.430588\pi$
$941$	13.2728	0.432682	0.216341	+	0.976318i	$0.430588\pi$
$942$	0	0
$943$	37.1572	1.21001
$944$	0	0
$945$	0.269662	0.00877211
$946$	0	0
$947$	−21.4809	−0.698034	−0.349017	−	0.937116i	$-0.613484\pi$
$947$	−21.4809	−0.698034	−0.349017	+	0.937116i	$0.613484\pi$
$948$	0	0
$949$	−52.8381	−1.71520
$950$	0	0
$951$	−32.9235	−1.06762
$952$	0	0
$953$	−21.0675	−0.682444	−0.341222	−	0.939983i	$-0.610841\pi$
$953$	−21.0675	−0.682444	−0.341222	+	0.939983i	$0.610841\pi$
$954$	0	0
$955$	−7.66041	−0.247885
$956$	0	0
$957$	3.62010	0.117021
$958$	0	0
$959$	−1.11337	−0.0359527
$960$	0	0
$961$	−16.3566	−0.527633
$962$	0	0
$963$	1.69311	0.0545598
$964$	0	0
$965$	8.81989	0.283922
$966$	0	0
$967$	15.6348	0.502782	0.251391	−	0.967886i	$-0.419112\pi$
$967$	15.6348	0.502782	0.251391	+	0.967886i	$0.419112\pi$
$968$	0	0
$969$	−7.83520	−0.251703
$970$	0	0
$971$	23.6485	0.758917	0.379458	−	0.925209i	$-0.376110\pi$
$971$	23.6485	0.758917	0.379458	+	0.925209i	$0.376110\pi$
$972$	0	0
$973$	−2.98540	−0.0957075
$974$	0	0
$975$	−60.5428	−1.93892
$976$	0	0
$977$	−8.41540	−0.269233	−0.134616	−	0.990898i	$-0.542980\pi$
$977$	−8.41540	−0.269233	−0.134616	+	0.990898i	$0.542980\pi$
$978$	0	0
$979$	−48.5066	−1.55028
$980$	0	0
$981$	15.9167	0.508181
$982$	0	0
$983$	−22.2133	−0.708495	−0.354247	−	0.935152i	$-0.615263\pi$
$983$	−22.2133	−0.708495	−0.354247	+	0.935152i	$0.615263\pi$
$984$	0	0
$985$	−6.69479	−0.213314
$986$	0	0
$987$	−2.27919	−0.0725474
$988$	0	0
$989$	22.1656	0.704824
$990$	0	0
$991$	24.2430	0.770104	0.385052	−	0.922895i	$-0.374184\pi$
$991$	24.2430	0.770104	0.385052	+	0.922895i	$0.374184\pi$
$992$	0	0
$993$	13.2282	0.419784
$994$	0	0
$995$	5.67685	0.179968
$996$	0	0
$997$	−16.0965	−0.509780	−0.254890	−	0.966970i	$-0.582039\pi$
$997$	−16.0965	−0.509780	−0.254890	+	0.966970i	$0.582039\pi$
$998$	0	0
$999$	10.1588	0.321410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.f.1.16	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.f.1.16	✓	19	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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30353 q^{3} +0.631218 q^{5} -0.267329 q^{7} +2.30625 q^{9} +O(q^{10})\)	\(q+2.30353 q^{3} +0.631218 q^{5} -0.267329 q^{7} +2.30625 q^{9} +3.34594 q^{11} +5.71167 q^{13} +1.45403 q^{15} +3.40139 q^{17} -1.00000 q^{19} -0.615800 q^{21} +3.34091 q^{23} -4.60156 q^{25} -1.59807 q^{27} +0.469687 q^{29} -3.82667 q^{31} +7.70748 q^{33} -0.168743 q^{35} -6.35693 q^{37} +13.1570 q^{39} +11.1219 q^{41} +6.63458 q^{43} +1.45575 q^{45} +3.70119 q^{47} -6.92854 q^{49} +7.83520 q^{51} -1.00000 q^{53} +2.11202 q^{55} -2.30353 q^{57} +0.818467 q^{59} +1.27038 q^{61} -0.616528 q^{63} +3.60531 q^{65} -9.76418 q^{67} +7.69590 q^{69} -4.93177 q^{71} -9.25090 q^{73} -10.5998 q^{75} -0.894467 q^{77} +4.83133 q^{79} -10.6000 q^{81} +7.98855 q^{83} +2.14702 q^{85} +1.08194 q^{87} -14.4971 q^{89} -1.52689 q^{91} -8.81484 q^{93} -0.631218 q^{95} +12.1173 q^{97} +7.71659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9	\( 19 q + 4 q^{3} + 4 q^{5} + 13 q^{7} + 19 q^{9} + q^{11} - q^{13} + 8 q^{15} + 3 q^{17} - 19 q^{19} + 8 q^{21} + 10 q^{23} + 21 q^{25} + 28 q^{27} + 2 q^{29} + 25 q^{31} + q^{33} + 20 q^{35} + 19 q^{37} + 37 q^{39} - 9 q^{41} + 35 q^{43} + 37 q^{45} + 23 q^{47} + 30 q^{49} + 34 q^{51} - 19 q^{53} + 40 q^{55} - 4 q^{57} + 16 q^{59} + 21 q^{61} + 3 q^{63} - 10 q^{65} + 67 q^{67} + 23 q^{69} + 18 q^{71} - 20 q^{73} + 33 q^{75} + 37 q^{77} + 2 q^{79} + 23 q^{81} + 38 q^{83} + 8 q^{85} + 18 q^{87} - q^{89} - 9 q^{91} + 14 q^{93} - 4 q^{95} - 21 q^{97} + 33 q^{99}+O(q^{100}) \) 19 * q + 4 * q^3 + 4 * q^5 + 13 * q^7 + 19 * q^9 + q^11 - q^13 + 8 * q^15 + 3 * q^17 - 19 * q^19 + 8 * q^21 + 10 * q^23 + 21 * q^25 + 28 * q^27 + 2 * q^29 + 25 * q^31 + q^33 + 20 * q^35 + 19 * q^37 + 37 * q^39 - 9 * q^41 + 35 * q^43 + 37 * q^45 + 23 * q^47 + 30 * q^49 + 34 * q^51 - 19 * q^53 + 40 * q^55 - 4 * q^57 + 16 * q^59 + 21 * q^61 + 3 * q^63 - 10 * q^65 + 67 * q^67 + 23 * q^69 + 18 * q^71 - 20 * q^73 + 33 * q^75 + 37 * q^77 + 2 * q^79 + 23 * q^81 + 38 * q^83 + 8 * q^85 + 18 * q^87 - q^89 - 9 * q^91 + 14 * q^93 - 4 * q^95 - 21 * q^97 + 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more