LMFDB - Embedded newform 4028.2.a.e.1.14

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} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + \cdots - 4 $ x^19 - 3x^18 - 35x^17 + 103x^16 + 501x^15 - 1437x^14 - 3775x^13 + 10450x^12 + 16076x^11 - 42255x^10 - 38701x^9 + 93907x^8 + 49522x^7 - 106284x^6 - 26703x^5 + 50522x^4 - 446x^3 - 6265x^2 + 719x - 4
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			1.14
Root			$1.68826$ 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)

$f(q)$	$=$	$q+1.68826 q^{3} +3.67694 q^{5} -0.618763 q^{7} -0.149789 q^{9} +O(q^{10})$	$q+1.68826 q^{3} +3.67694 q^{5} -0.618763 q^{7} -0.149789 q^{9} +1.41023 q^{11} +1.19056 q^{13} +6.20763 q^{15} +1.38613 q^{17} +1.00000 q^{19} -1.04463 q^{21} +5.40402 q^{23} +8.51991 q^{25} -5.31765 q^{27} +5.27827 q^{29} +5.74573 q^{31} +2.38082 q^{33} -2.27516 q^{35} -8.98475 q^{37} +2.00997 q^{39} -1.50551 q^{41} +2.14281 q^{43} -0.550765 q^{45} -11.4142 q^{47} -6.61713 q^{49} +2.34014 q^{51} +1.00000 q^{53} +5.18532 q^{55} +1.68826 q^{57} +10.5577 q^{59} +1.35643 q^{61} +0.0926839 q^{63} +4.37761 q^{65} -5.35599 q^{67} +9.12337 q^{69} +3.52908 q^{71} +4.19448 q^{73} +14.3838 q^{75} -0.872596 q^{77} -10.5434 q^{79} -8.52820 q^{81} +14.9277 q^{83} +5.09672 q^{85} +8.91107 q^{87} +2.74302 q^{89} -0.736673 q^{91} +9.70027 q^{93} +3.67694 q^{95} -0.00820032 q^{97} -0.211236 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * 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
$3$	1.68826	0.974716	0.487358	−	0.873202i	$-0.337961\pi$
$3$	1.68826	0.974716	0.487358	+	0.873202i	$0.337961\pi$
$4$	0	0
$5$	3.67694	1.64438	0.822190	−	0.569214i	$-0.192752\pi$
$5$	3.67694	1.64438	0.822190	+	0.569214i	$0.192752\pi$
$6$	0	0
$7$	−0.618763	−0.233871	−0.116935	−	0.993140i	$-0.537307\pi$
$7$	−0.618763	−0.233871	−0.116935	+	0.993140i	$0.537307\pi$
$8$	0	0
$9$	−0.149789	−0.0499296
$10$	0	0
$11$	1.41023	0.425199	0.212600	−	0.977139i	$-0.431807\pi$
$11$	1.41023	0.425199	0.212600	+	0.977139i	$0.431807\pi$
$12$	0	0
$13$	1.19056	0.330201	0.165101	−	0.986277i	$-0.447205\pi$
$13$	1.19056	0.330201	0.165101	+	0.986277i	$0.447205\pi$
$14$	0	0
$15$	6.20763	1.60280
$16$	0	0
$17$	1.38613	0.336186	0.168093	−	0.985771i	$-0.446239\pi$
$17$	1.38613	0.336186	0.168093	+	0.985771i	$0.446239\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.04463	−0.227957
$22$	0	0
$23$	5.40402	1.12682	0.563408	−	0.826179i	$-0.309490\pi$
$23$	5.40402	1.12682	0.563408	+	0.826179i	$0.309490\pi$
$24$	0	0
$25$	8.51991	1.70398
$26$	0	0
$27$	−5.31765	−1.02338
$28$	0	0
$29$	5.27827	0.980150	0.490075	−	0.871680i	$-0.336969\pi$
$29$	5.27827	0.980150	0.490075	+	0.871680i	$0.336969\pi$
$30$	0	0
$31$	5.74573	1.03196	0.515982	−	0.856600i	$-0.327427\pi$
$31$	5.74573	1.03196	0.515982	+	0.856600i	$0.327427\pi$
$32$	0	0
$33$	2.38082	0.414448
$34$	0	0
$35$	−2.27516	−0.384572
$36$	0	0
$37$	−8.98475	−1.47708	−0.738542	−	0.674207i	$-0.764485\pi$
$37$	−8.98475	−1.47708	−0.738542	+	0.674207i	$0.764485\pi$
$38$	0	0
$39$	2.00997	0.321852
$40$	0	0
$41$	−1.50551	−0.235121	−0.117560	−	0.993066i	$-0.537507\pi$
$41$	−1.50551	−0.235121	−0.117560	+	0.993066i	$0.537507\pi$
$42$	0	0
$43$	2.14281	0.326776	0.163388	−	0.986562i	$-0.447758\pi$
$43$	2.14281	0.326776	0.163388	+	0.986562i	$0.447758\pi$
$44$	0	0
$45$	−0.550765	−0.0821032
$46$	0	0
$47$	−11.4142	−1.66494	−0.832468	−	0.554072i	$-0.813073\pi$
$47$	−11.4142	−1.66494	−0.832468	+	0.554072i	$0.813073\pi$
$48$	0	0
$49$	−6.61713	−0.945305
$50$	0	0
$51$	2.34014	0.327685
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	5.18532	0.699189
$56$	0	0
$57$	1.68826	0.223615
$58$	0	0
$59$	10.5577	1.37449	0.687246	−	0.726425i	$-0.258819\pi$
$59$	10.5577	1.37449	0.687246	+	0.726425i	$0.258819\pi$
$60$	0	0
$61$	1.35643	0.173673	0.0868366	−	0.996223i	$-0.472324\pi$
$61$	1.35643	0.173673	0.0868366	+	0.996223i	$0.472324\pi$
$62$	0	0
$63$	0.0926839	0.0116771
$64$	0	0
$65$	4.37761	0.542976
$66$	0	0
$67$	−5.35599	−0.654339	−0.327169	−	0.944966i	$-0.606095\pi$
$67$	−5.35599	−0.654339	−0.327169	+	0.944966i	$0.606095\pi$
$68$	0	0
$69$	9.12337	1.09832
$70$	0	0
$71$	3.52908	0.418825	0.209413	−	0.977827i	$-0.432845\pi$
$71$	3.52908	0.418825	0.209413	+	0.977827i	$0.432845\pi$
$72$	0	0
$73$	4.19448	0.490927	0.245464	−	0.969406i	$-0.421060\pi$
$73$	4.19448	0.490927	0.245464	+	0.969406i	$0.421060\pi$
$74$	0	0
$75$	14.3838	1.66090
$76$	0	0
$77$	−0.872596	−0.0994416
$78$	0	0
$79$	−10.5434	−1.18622	−0.593110	−	0.805121i	$-0.702100\pi$
$79$	−10.5434	−1.18622	−0.593110	+	0.805121i	$0.702100\pi$
$80$	0	0
$81$	−8.52820	−0.947577
$82$	0	0
$83$	14.9277	1.63853	0.819265	−	0.573416i	$-0.194382\pi$
$83$	14.9277	1.63853	0.819265	+	0.573416i	$0.194382\pi$
$84$	0	0
$85$	5.09672	0.552817
$86$	0	0
$87$	8.91107	0.955367
$88$	0	0
$89$	2.74302	0.290760	0.145380	−	0.989376i	$-0.453560\pi$
$89$	2.74302	0.290760	0.145380	+	0.989376i	$0.453560\pi$
$90$	0	0
$91$	−0.736673	−0.0772243
$92$	0	0
$93$	9.70027	1.00587
$94$	0	0
$95$	3.67694	0.377246
$96$	0	0
$97$	−0.00820032	−0.000832616 0	−0.000416308	−	1.00000i	$-0.500133\pi$
$97$	−0.00820032	−0.000832616 0	−0.000416308		1.00000i	$0.500133\pi$
$98$	0	0
$99$	−0.211236	−0.0212300
$100$	0	0
$101$	−1.23046	−0.122435	−0.0612177	−	0.998124i	$-0.519498\pi$
$101$	−1.23046	−0.122435	−0.0612177	+	0.998124i	$0.519498\pi$
$102$	0	0
$103$	12.6840	1.24979	0.624894	−	0.780709i	$-0.285142\pi$
$103$	12.6840	1.24979	0.624894	+	0.780709i	$0.285142\pi$
$104$	0	0
$105$	−3.84105	−0.374848
$106$	0	0
$107$	−4.15631	−0.401806	−0.200903	−	0.979611i	$-0.564388\pi$
$107$	−4.15631	−0.401806	−0.200903	+	0.979611i	$0.564388\pi$
$108$	0	0
$109$	3.23082	0.309457	0.154728	−	0.987957i	$-0.450550\pi$
$109$	3.23082	0.309457	0.154728	+	0.987957i	$0.450550\pi$
$110$	0	0
$111$	−15.1686	−1.43974
$112$	0	0
$113$	11.2163	1.05514	0.527572	−	0.849510i	$-0.323102\pi$
$113$	11.2163	1.05514	0.527572	+	0.849510i	$0.323102\pi$
$114$	0	0
$115$	19.8703	1.85291
$116$	0	0
$117$	−0.178332	−0.0164868
$118$	0	0
$119$	−0.857686	−0.0786239
$120$	0	0
$121$	−9.01126	−0.819206
$122$	0	0
$123$	−2.54168	−0.229176
$124$	0	0
$125$	12.9425	1.15761
$126$	0	0
$127$	0.712277	0.0632044	0.0316022	−	0.999501i	$-0.489939\pi$
$127$	0.712277	0.0632044	0.0316022	+	0.999501i	$0.489939\pi$
$128$	0	0
$129$	3.61762	0.318514
$130$	0	0
$131$	−6.19447	−0.541213	−0.270607	−	0.962690i	$-0.587224\pi$
$131$	−6.19447	−0.541213	−0.270607	+	0.962690i	$0.587224\pi$
$132$	0	0
$133$	−0.618763	−0.0536536
$134$	0	0
$135$	−19.5527	−1.68283
$136$	0	0
$137$	−11.9736	−1.02297	−0.511485	−	0.859292i	$-0.670904\pi$
$137$	−11.9736	−1.02297	−0.511485	+	0.859292i	$0.670904\pi$
$138$	0	0
$139$	13.7031	1.16228	0.581140	−	0.813803i	$-0.302607\pi$
$139$	13.7031	1.16228	0.581140	+	0.813803i	$0.302607\pi$
$140$	0	0
$141$	−19.2702	−1.62284
$142$	0	0
$143$	1.67896	0.140401
$144$	0	0
$145$	19.4079	1.61174
$146$	0	0
$147$	−11.1714	−0.921403
$148$	0	0
$149$	−22.7297	−1.86209	−0.931046	−	0.364901i	$-0.881103\pi$
$149$	−22.7297	−1.86209	−0.931046	+	0.364901i	$0.881103\pi$
$150$	0	0
$151$	−19.9503	−1.62353	−0.811767	−	0.583982i	$-0.801494\pi$
$151$	−19.9503	−1.62353	−0.811767	+	0.583982i	$0.801494\pi$
$152$	0	0
$153$	−0.207627	−0.0167856
$154$	0	0
$155$	21.1267	1.69694
$156$	0	0
$157$	8.04530	0.642085	0.321042	−	0.947065i	$-0.395967\pi$
$157$	8.04530	0.642085	0.321042	+	0.947065i	$0.395967\pi$
$158$	0	0
$159$	1.68826	0.133887
$160$	0	0
$161$	−3.34381	−0.263529
$162$	0	0
$163$	6.13251	0.480335	0.240168	−	0.970731i	$-0.422798\pi$
$163$	6.13251	0.480335	0.240168	+	0.970731i	$0.422798\pi$
$164$	0	0
$165$	8.75416	0.681510
$166$	0	0
$167$	9.55584	0.739453	0.369727	−	0.929141i	$-0.379451\pi$
$167$	9.55584	0.739453	0.369727	+	0.929141i	$0.379451\pi$
$168$	0	0
$169$	−11.5826	−0.890967
$170$	0	0
$171$	−0.149789	−0.0114546
$172$	0	0
$173$	−6.15328	−0.467826	−0.233913	−	0.972258i	$-0.575153\pi$
$173$	−6.15328	−0.467826	−0.233913	+	0.972258i	$0.575153\pi$
$174$	0	0
$175$	−5.27181	−0.398511
$176$	0	0
$177$	17.8241	1.33974
$178$	0	0
$179$	7.24559	0.541561	0.270780	−	0.962641i	$-0.412718\pi$
$179$	7.24559	0.541561	0.270780	+	0.962641i	$0.412718\pi$
$180$	0	0
$181$	15.4535	1.14865	0.574323	−	0.818629i	$-0.305265\pi$
$181$	15.4535	1.14865	0.574323	+	0.818629i	$0.305265\pi$
$182$	0	0
$183$	2.29000	0.169282
$184$	0	0
$185$	−33.0364	−2.42889
$186$	0	0
$187$	1.95476	0.142946
$188$	0	0
$189$	3.29037	0.239339
$190$	0	0
$191$	23.2659	1.68346	0.841729	−	0.539900i	$-0.181538\pi$
$191$	23.2659	1.68346	0.841729	+	0.539900i	$0.181538\pi$
$192$	0	0
$193$	−12.2297	−0.880316	−0.440158	−	0.897920i	$-0.645078\pi$
$193$	−12.2297	−0.880316	−0.440158	+	0.897920i	$0.645078\pi$
$194$	0	0
$195$	7.39053	0.529247
$196$	0	0
$197$	15.4497	1.10074	0.550372	−	0.834920i	$-0.314486\pi$
$197$	15.4497	1.10074	0.550372	+	0.834920i	$0.314486\pi$
$198$	0	0
$199$	−21.3236	−1.51159	−0.755795	−	0.654808i	$-0.772750\pi$
$199$	−21.3236	−1.51159	−0.755795	+	0.654808i	$0.772750\pi$
$200$	0	0
$201$	−9.04229	−0.637794
$202$	0	0
$203$	−3.26600	−0.229228
$204$	0	0
$205$	−5.53567	−0.386628
$206$	0	0
$207$	−0.809462	−0.0562615
$208$	0	0
$209$	1.41023	0.0975474
$210$	0	0
$211$	1.71839	0.118299	0.0591494	−	0.998249i	$-0.481161\pi$
$211$	1.71839	0.118299	0.0591494	+	0.998249i	$0.481161\pi$
$212$	0	0
$213$	5.95800	0.408236
$214$	0	0
$215$	7.87901	0.537344
$216$	0	0
$217$	−3.55525	−0.241346
$218$	0	0
$219$	7.08136	0.478514
$220$	0	0
$221$	1.65027	0.111009
$222$	0	0
$223$	3.45506	0.231368	0.115684	−	0.993286i	$-0.463094\pi$
$223$	3.45506	0.231368	0.115684	+	0.993286i	$0.463094\pi$
$224$	0	0
$225$	−1.27619	−0.0850792
$226$	0	0
$227$	12.8362	0.851971	0.425986	−	0.904730i	$-0.359927\pi$
$227$	12.8362	0.851971	0.425986	+	0.904730i	$0.359927\pi$
$228$	0	0
$229$	−26.0424	−1.72093	−0.860465	−	0.509510i	$-0.829827\pi$
$229$	−26.0424	−1.72093	−0.860465	+	0.509510i	$0.829827\pi$
$230$	0	0
$231$	−1.47317	−0.0969272
$232$	0	0
$233$	17.8557	1.16977	0.584884	−	0.811117i	$-0.301140\pi$
$233$	17.8557	1.16977	0.584884	+	0.811117i	$0.301140\pi$
$234$	0	0
$235$	−41.9695	−2.73779
$236$	0	0
$237$	−17.7999	−1.15623
$238$	0	0
$239$	29.2352	1.89107	0.945534	−	0.325522i	$-0.105540\pi$
$239$	29.2352	1.89107	0.945534	+	0.325522i	$0.105540\pi$
$240$	0	0
$241$	2.75212	0.177279	0.0886397	−	0.996064i	$-0.471748\pi$
$241$	2.75212	0.177279	0.0886397	+	0.996064i	$0.471748\pi$
$242$	0	0
$243$	1.55517	0.0997643
$244$	0	0
$245$	−24.3308	−1.55444
$246$	0	0
$247$	1.19056	0.0757534
$248$	0	0
$249$	25.2018	1.59710
$250$	0	0
$251$	−26.9696	−1.70231	−0.851154	−	0.524916i	$-0.824097\pi$
$251$	−26.9696	−1.70231	−0.851154	+	0.524916i	$0.824097\pi$
$252$	0	0
$253$	7.62089	0.479121
$254$	0	0
$255$	8.60457	0.538839
$256$	0	0
$257$	6.00307	0.374461	0.187231	−	0.982316i	$-0.440049\pi$
$257$	6.00307	0.374461	0.187231	+	0.982316i	$0.440049\pi$
$258$	0	0
$259$	5.55944	0.345447
$260$	0	0
$261$	−0.790626	−0.0489385
$262$	0	0
$263$	−3.95181	−0.243679	−0.121840	−	0.992550i	$-0.538879\pi$
$263$	−3.95181	−0.243679	−0.121840	+	0.992550i	$0.538879\pi$
$264$	0	0
$265$	3.67694	0.225873
$266$	0	0
$267$	4.63092	0.283408
$268$	0	0
$269$	−20.1323	−1.22749	−0.613743	−	0.789506i	$-0.710337\pi$
$269$	−20.1323	−1.22749	−0.613743	+	0.789506i	$0.710337\pi$
$270$	0	0
$271$	−16.2475	−0.986966	−0.493483	−	0.869756i	$-0.664276\pi$
$271$	−16.2475	−0.986966	−0.493483	+	0.869756i	$0.664276\pi$
$272$	0	0
$273$	−1.24369	−0.0752718
$274$	0	0
$275$	12.0150	0.724532
$276$	0	0
$277$	−18.6166	−1.11856	−0.559281	−	0.828978i	$-0.688923\pi$
$277$	−18.6166	−1.11856	−0.559281	+	0.828978i	$0.688923\pi$
$278$	0	0
$279$	−0.860647	−0.0515256
$280$	0	0
$281$	−7.19987	−0.429508	−0.214754	−	0.976668i	$-0.568895\pi$
$281$	−7.19987	−0.429508	−0.214754	+	0.976668i	$0.568895\pi$
$282$	0	0
$283$	−25.7244	−1.52916	−0.764579	−	0.644530i	$-0.777053\pi$
$283$	−25.7244	−1.52916	−0.764579	+	0.644530i	$0.777053\pi$
$284$	0	0
$285$	6.20763	0.367708
$286$	0	0
$287$	0.931553	0.0549879
$288$	0	0
$289$	−15.0786	−0.886979
$290$	0	0
$291$	−0.0138442	−0.000811564 0
$292$	0	0
$293$	−18.4599	−1.07844	−0.539218	−	0.842166i	$-0.681280\pi$
$293$	−18.4599	−1.07844	−0.539218	+	0.842166i	$0.681280\pi$
$294$	0	0
$295$	38.8200	2.26019
$296$	0	0
$297$	−7.49909	−0.435142
$298$	0	0
$299$	6.43379	0.372076
$300$	0	0
$301$	−1.32589	−0.0764233
$302$	0	0
$303$	−2.07733	−0.119340
$304$	0	0
$305$	4.98752	0.285584
$306$	0	0
$307$	−26.7751	−1.52814	−0.764068	−	0.645136i	$-0.776801\pi$
$307$	−26.7751	−1.52814	−0.764068	+	0.645136i	$0.776801\pi$
$308$	0	0
$309$	21.4138	1.21819
$310$	0	0
$311$	−10.8269	−0.613936	−0.306968	−	0.951720i	$-0.599314\pi$
$311$	−10.8269	−0.613936	−0.306968	+	0.951720i	$0.599314\pi$
$312$	0	0
$313$	−15.5028	−0.876271	−0.438136	−	0.898909i	$-0.644361\pi$
$313$	−15.5028	−0.876271	−0.438136	+	0.898909i	$0.644361\pi$
$314$	0	0
$315$	0.340793	0.0192015
$316$	0	0
$317$	8.52849	0.479008	0.239504	−	0.970895i	$-0.423015\pi$
$317$	8.52849	0.479008	0.239504	+	0.970895i	$0.423015\pi$
$318$	0	0
$319$	7.44355	0.416759
$320$	0	0
$321$	−7.01693	−0.391647
$322$	0	0
$323$	1.38613	0.0771263
$324$	0	0
$325$	10.1434	0.562657
$326$	0	0
$327$	5.45446	0.301632
$328$	0	0
$329$	7.06271	0.389380
$330$	0	0
$331$	33.1811	1.82380	0.911898	−	0.410417i	$-0.134617\pi$
$331$	33.1811	1.82380	0.911898	+	0.410417i	$0.134617\pi$
$332$	0	0
$333$	1.34582	0.0737503
$334$	0	0
$335$	−19.6937	−1.07598
$336$	0	0
$337$	−27.8653	−1.51792	−0.758959	−	0.651138i	$-0.774292\pi$
$337$	−27.8653	−1.51792	−0.758959	+	0.651138i	$0.774292\pi$
$338$	0	0
$339$	18.9361	1.02847
$340$	0	0
$341$	8.10278	0.438790
$342$	0	0
$343$	8.42578	0.454949
$344$	0	0
$345$	33.5461	1.80606
$346$	0	0
$347$	21.4425	1.15109	0.575546	−	0.817769i	$-0.304789\pi$
$347$	21.4425	1.15109	0.575546	+	0.817769i	$0.304789\pi$
$348$	0	0
$349$	−15.3668	−0.822565	−0.411283	−	0.911508i	$-0.634919\pi$
$349$	−15.3668	−0.822565	−0.411283	+	0.911508i	$0.634919\pi$
$350$	0	0
$351$	−6.33097	−0.337922
$352$	0	0
$353$	3.27639	0.174385	0.0871923	−	0.996191i	$-0.472211\pi$
$353$	3.27639	0.174385	0.0871923	+	0.996191i	$0.472211\pi$
$354$	0	0
$355$	12.9762	0.688708
$356$	0	0
$357$	−1.44799	−0.0766360
$358$	0	0
$359$	−15.8090	−0.834367	−0.417183	−	0.908822i	$-0.636983\pi$
$359$	−15.8090	−0.834367	−0.417183	+	0.908822i	$0.636983\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−15.2133	−0.798492
$364$	0	0
$365$	15.4229	0.807270
$366$	0	0
$367$	9.95675	0.519738	0.259869	−	0.965644i	$-0.416321\pi$
$367$	9.95675	0.519738	0.259869	+	0.965644i	$0.416321\pi$
$368$	0	0
$369$	0.225508	0.0117395
$370$	0	0
$371$	−0.618763	−0.0321246
$372$	0	0
$373$	−25.2933	−1.30964	−0.654819	−	0.755785i	$-0.727255\pi$
$373$	−25.2933	−1.30964	−0.654819	+	0.755785i	$0.727255\pi$
$374$	0	0
$375$	21.8503	1.12834
$376$	0	0
$377$	6.28408	0.323647
$378$	0	0
$379$	18.5412	0.952400	0.476200	−	0.879337i	$-0.342014\pi$
$379$	18.5412	0.952400	0.476200	+	0.879337i	$0.342014\pi$
$380$	0	0
$381$	1.20251	0.0616063
$382$	0	0
$383$	11.2072	0.572663	0.286332	−	0.958131i	$-0.407564\pi$
$383$	11.2072	0.572663	0.286332	+	0.958131i	$0.407564\pi$
$384$	0	0
$385$	−3.20849	−0.163520
$386$	0	0
$387$	−0.320970	−0.0163158
$388$	0	0
$389$	−6.80618	−0.345087	−0.172544	−	0.985002i	$-0.555199\pi$
$389$	−6.80618	−0.345087	−0.172544	+	0.985002i	$0.555199\pi$
$390$	0	0
$391$	7.49067	0.378819
$392$	0	0
$393$	−10.4579	−0.527529
$394$	0	0
$395$	−38.7673	−1.95060
$396$	0	0
$397$	0.651836	0.0327147	0.0163573	−	0.999866i	$-0.494793\pi$
$397$	0.651836	0.0327147	0.0163573	+	0.999866i	$0.494793\pi$
$398$	0	0
$399$	−1.04463	−0.0522970
$400$	0	0
$401$	20.0608	1.00179	0.500895	−	0.865508i	$-0.333004\pi$
$401$	20.0608	1.00179	0.500895	+	0.865508i	$0.333004\pi$
$402$	0	0
$403$	6.84062	0.340756
$404$	0	0
$405$	−31.3577	−1.55818
$406$	0	0
$407$	−12.6705	−0.628055
$408$	0	0
$409$	−12.6064	−0.623348	−0.311674	−	0.950189i	$-0.600890\pi$
$409$	−12.6064	−0.623348	−0.311674	+	0.950189i	$0.600890\pi$
$410$	0	0
$411$	−20.2144	−0.997105
$412$	0	0
$413$	−6.53270	−0.321453
$414$	0	0
$415$	54.8883	2.69436
$416$	0	0
$417$	23.1343	1.13289
$418$	0	0
$419$	1.43940	0.0703194	0.0351597	−	0.999382i	$-0.488806\pi$
$419$	1.43940	0.0703194	0.0351597	+	0.999382i	$0.488806\pi$
$420$	0	0
$421$	30.9282	1.50735	0.753675	−	0.657247i	$-0.228279\pi$
$421$	30.9282	1.50735	0.753675	+	0.657247i	$0.228279\pi$
$422$	0	0
$423$	1.70973	0.0831297
$424$	0	0
$425$	11.8097	0.572855
$426$	0	0
$427$	−0.839309	−0.0406170
$428$	0	0
$429$	2.83451	0.136851
$430$	0	0
$431$	−5.45924	−0.262962	−0.131481	−	0.991319i	$-0.541973\pi$
$431$	−5.45924	−0.262962	−0.131481	+	0.991319i	$0.541973\pi$
$432$	0	0
$433$	29.5301	1.41913	0.709563	−	0.704642i	$-0.248893\pi$
$433$	29.5301	1.41913	0.709563	+	0.704642i	$0.248893\pi$
$434$	0	0
$435$	32.7655	1.57099
$436$	0	0
$437$	5.40402	0.258509
$438$	0	0
$439$	13.2569	0.632719	0.316360	−	0.948639i	$-0.397539\pi$
$439$	13.2569	0.632719	0.316360	+	0.948639i	$0.397539\pi$
$440$	0	0
$441$	0.991173	0.0471987
$442$	0	0
$443$	−12.7976	−0.608031	−0.304015	−	0.952667i	$-0.598327\pi$
$443$	−12.7976	−0.608031	−0.304015	+	0.952667i	$0.598327\pi$
$444$	0	0
$445$	10.0859	0.478119
$446$	0	0
$447$	−38.3736	−1.81501
$448$	0	0
$449$	−9.65458	−0.455628	−0.227814	−	0.973705i	$-0.573158\pi$
$449$	−9.65458	−0.455628	−0.227814	+	0.973705i	$0.573158\pi$
$450$	0	0
$451$	−2.12311	−0.0999732
$452$	0	0
$453$	−33.6813	−1.58248
$454$	0	0
$455$	−2.70871	−0.126986
$456$	0	0
$457$	19.6845	0.920804	0.460402	−	0.887710i	$-0.347705\pi$
$457$	19.6845	0.920804	0.460402	+	0.887710i	$0.347705\pi$
$458$	0	0
$459$	−7.37095	−0.344047
$460$	0	0
$461$	15.5956	0.726358	0.363179	−	0.931719i	$-0.381691\pi$
$461$	15.5956	0.726358	0.363179	+	0.931719i	$0.381691\pi$
$462$	0	0
$463$	12.5930	0.585248	0.292624	−	0.956228i	$-0.405472\pi$
$463$	12.5930	0.585248	0.292624	+	0.956228i	$0.405472\pi$
$464$	0	0
$465$	35.6673	1.65403
$466$	0	0
$467$	−20.3179	−0.940200	−0.470100	−	0.882613i	$-0.655782\pi$
$467$	−20.3179	−0.940200	−0.470100	+	0.882613i	$0.655782\pi$
$468$	0	0
$469$	3.31409	0.153031
$470$	0	0
$471$	13.5825	0.625850
$472$	0	0
$473$	3.02185	0.138945
$474$	0	0
$475$	8.51991	0.390920
$476$	0	0
$477$	−0.149789	−0.00685836
$478$	0	0
$479$	36.2144	1.65468	0.827339	−	0.561703i	$-0.189854\pi$
$479$	36.2144	1.65468	0.827339	+	0.561703i	$0.189854\pi$
$480$	0	0
$481$	−10.6969	−0.487735
$482$	0	0
$483$	−5.64521	−0.256866
$484$	0	0
$485$	−0.0301521	−0.00136914
$486$	0	0
$487$	−20.8111	−0.943040	−0.471520	−	0.881855i	$-0.656295\pi$
$487$	−20.8111	−0.943040	−0.471520	+	0.881855i	$0.656295\pi$
$488$	0	0
$489$	10.3533	0.468190
$490$	0	0
$491$	3.91285	0.176584	0.0882922	−	0.996095i	$-0.471859\pi$
$491$	3.91285	0.176584	0.0882922	+	0.996095i	$0.471859\pi$
$492$	0	0
$493$	7.31636	0.329512
$494$	0	0
$495$	−0.776704	−0.0349102
$496$	0	0
$497$	−2.18367	−0.0979509
$498$	0	0
$499$	−22.2153	−0.994495	−0.497247	−	0.867609i	$-0.665656\pi$
$499$	−22.2153	−0.994495	−0.497247	+	0.867609i	$0.665656\pi$
$500$	0	0
$501$	16.1327	0.720757
$502$	0	0
$503$	14.0351	0.625796	0.312898	−	0.949787i	$-0.398700\pi$
$503$	14.0351	0.625796	0.312898	+	0.949787i	$0.398700\pi$
$504$	0	0
$505$	−4.52433	−0.201330
$506$	0	0
$507$	−19.5544	−0.868440
$508$	0	0
$509$	−10.5464	−0.467463	−0.233731	−	0.972301i	$-0.575094\pi$
$509$	−10.5464	−0.467463	−0.233731	+	0.972301i	$0.575094\pi$
$510$	0	0
$511$	−2.59539	−0.114813
$512$	0	0
$513$	−5.31765	−0.234780
$514$	0	0
$515$	46.6382	2.05513
$516$	0	0
$517$	−16.0967	−0.707930
$518$	0	0
$519$	−10.3883	−0.455997
$520$	0	0
$521$	24.0280	1.05269	0.526343	−	0.850273i	$-0.323563\pi$
$521$	24.0280	1.05269	0.526343	+	0.850273i	$0.323563\pi$
$522$	0	0
$523$	−9.92937	−0.434181	−0.217091	−	0.976151i	$-0.569657\pi$
$523$	−9.92937	−0.434181	−0.217091	+	0.976151i	$0.569657\pi$
$524$	0	0
$525$	−8.90017	−0.388435
$526$	0	0
$527$	7.96432	0.346931
$528$	0	0
$529$	6.20341	0.269714
$530$	0	0
$531$	−1.58142	−0.0686279
$532$	0	0
$533$	−1.79239	−0.0776372
$534$	0	0
$535$	−15.2825	−0.660722
$536$	0	0
$537$	12.2324	0.527868
$538$	0	0
$539$	−9.33165	−0.401943
$540$	0	0
$541$	−24.6043	−1.05782	−0.528910	−	0.848678i	$-0.677399\pi$
$541$	−24.6043	−1.05782	−0.528910	+	0.848678i	$0.677399\pi$
$542$	0	0
$543$	26.0894	1.11960
$544$	0	0
$545$	11.8795	0.508864
$546$	0	0
$547$	−36.3643	−1.55483	−0.777413	−	0.628990i	$-0.783469\pi$
$547$	−36.3643	−1.55483	−0.777413	+	0.628990i	$0.783469\pi$
$548$	0	0
$549$	−0.203178	−0.00867143
$550$	0	0
$551$	5.27827	0.224862
$552$	0	0
$553$	6.52384	0.277422
$554$	0	0
$555$	−55.7740	−2.36747
$556$	0	0
$557$	−32.7814	−1.38899	−0.694496	−	0.719497i	$-0.744373\pi$
$557$	−32.7814	−1.38899	−0.694496	+	0.719497i	$0.744373\pi$
$558$	0	0
$559$	2.55114	0.107902
$560$	0	0
$561$	3.30013	0.139332
$562$	0	0
$563$	−42.0911	−1.77393	−0.886965	−	0.461837i	$-0.847191\pi$
$563$	−42.0911	−1.77393	−0.886965	+	0.461837i	$0.847191\pi$
$564$	0	0
$565$	41.2419	1.73506
$566$	0	0
$567$	5.27693	0.221610
$568$	0	0
$569$	23.1641	0.971091	0.485546	−	0.874211i	$-0.338621\pi$
$569$	23.1641	0.971091	0.485546	+	0.874211i	$0.338621\pi$
$570$	0	0
$571$	−33.0160	−1.38168	−0.690839	−	0.723009i	$-0.742759\pi$
$571$	−33.0160	−1.38168	−0.690839	+	0.723009i	$0.742759\pi$
$572$	0	0
$573$	39.2787	1.64089
$574$	0	0
$575$	46.0418	1.92007
$576$	0	0
$577$	11.4900	0.478334	0.239167	−	0.970978i	$-0.423126\pi$
$577$	11.4900	0.478334	0.239167	+	0.970978i	$0.423126\pi$
$578$	0	0
$579$	−20.6469	−0.858058
$580$	0	0
$581$	−9.23672	−0.383204
$582$	0	0
$583$	1.41023	0.0584056
$584$	0	0
$585$	−0.655718	−0.0271106
$586$	0	0
$587$	−28.7302	−1.18582	−0.592912	−	0.805267i	$-0.702022\pi$
$587$	−28.7302	−1.18582	−0.592912	+	0.805267i	$0.702022\pi$
$588$	0	0
$589$	5.74573	0.236749
$590$	0	0
$591$	26.0830	1.07291
$592$	0	0
$593$	−37.8874	−1.55585	−0.777925	−	0.628357i	$-0.783728\pi$
$593$	−37.8874	−1.55585	−0.777925	+	0.628357i	$0.783728\pi$
$594$	0	0
$595$	−3.15366	−0.129288
$596$	0	0
$597$	−35.9997	−1.47337
$598$	0	0
$599$	−22.4615	−0.917752	−0.458876	−	0.888500i	$-0.651748\pi$
$599$	−22.4615	−0.917752	−0.458876	+	0.888500i	$0.651748\pi$
$600$	0	0
$601$	39.3850	1.60655	0.803274	−	0.595610i	$-0.203090\pi$
$601$	39.3850	1.60655	0.803274	+	0.595610i	$0.203090\pi$
$602$	0	0
$603$	0.802268	0.0326709
$604$	0	0
$605$	−33.1339	−1.34708
$606$	0	0
$607$	6.57162	0.266734	0.133367	−	0.991067i	$-0.457421\pi$
$607$	6.57162	0.266734	0.133367	+	0.991067i	$0.457421\pi$
$608$	0	0
$609$	−5.51384	−0.223432
$610$	0	0
$611$	−13.5893	−0.549764
$612$	0	0
$613$	18.9365	0.764840	0.382420	−	0.923989i	$-0.375091\pi$
$613$	18.9365	0.764840	0.382420	+	0.923989i	$0.375091\pi$
$614$	0	0
$615$	−9.34563	−0.376852
$616$	0	0
$617$	12.3311	0.496432	0.248216	−	0.968705i	$-0.420156\pi$
$617$	12.3311	0.496432	0.248216	+	0.968705i	$0.420156\pi$
$618$	0	0
$619$	42.5986	1.71218	0.856091	−	0.516824i	$-0.172886\pi$
$619$	42.5986	1.71218	0.856091	+	0.516824i	$0.172886\pi$
$620$	0	0
$621$	−28.7367	−1.15316
$622$	0	0
$623$	−1.69728	−0.0680001
$624$	0	0
$625$	4.98937	0.199575
$626$	0	0
$627$	2.38082	0.0950810
$628$	0	0
$629$	−12.4540	−0.496575
$630$	0	0
$631$	−21.0644	−0.838561	−0.419281	−	0.907857i	$-0.637718\pi$
$631$	−21.0644	−0.838561	−0.419281	+	0.907857i	$0.637718\pi$
$632$	0	0
$633$	2.90108	0.115308
$634$	0	0
$635$	2.61900	0.103932
$636$	0	0
$637$	−7.87808	−0.312141
$638$	0	0
$639$	−0.528618	−0.0209118
$640$	0	0
$641$	−45.8668	−1.81163	−0.905815	−	0.423673i	$-0.860741\pi$
$641$	−45.8668	−1.81163	−0.905815	+	0.423673i	$0.860741\pi$
$642$	0	0
$643$	−16.8300	−0.663708	−0.331854	−	0.943331i	$-0.607674\pi$
$643$	−16.8300	−0.663708	−0.331854	+	0.943331i	$0.607674\pi$
$644$	0	0
$645$	13.3018	0.523757
$646$	0	0
$647$	27.3678	1.07594	0.537970	−	0.842964i	$-0.319191\pi$
$647$	27.3678	1.07594	0.537970	+	0.842964i	$0.319191\pi$
$648$	0	0
$649$	14.8887	0.584433
$650$	0	0
$651$	−6.00217	−0.235244
$652$	0	0
$653$	−18.9826	−0.742847	−0.371423	−	0.928464i	$-0.621130\pi$
$653$	−18.9826	−0.742847	−0.371423	+	0.928464i	$0.621130\pi$
$654$	0	0
$655$	−22.7767	−0.889960
$656$	0	0
$657$	−0.628287	−0.0245118
$658$	0	0
$659$	−21.1664	−0.824525	−0.412262	−	0.911065i	$-0.635261\pi$
$659$	−21.1664	−0.824525	−0.412262	+	0.911065i	$0.635261\pi$
$660$	0	0
$661$	1.03035	0.0400760	0.0200380	−	0.999799i	$-0.493621\pi$
$661$	1.03035	0.0400760	0.0200380	+	0.999799i	$0.493621\pi$
$662$	0	0
$663$	2.78607	0.108202
$664$	0	0
$665$	−2.27516	−0.0882268
$666$	0	0
$667$	28.5239	1.10445
$668$	0	0
$669$	5.83303	0.225518
$670$	0	0
$671$	1.91287	0.0738457
$672$	0	0
$673$	−41.2551	−1.59027	−0.795133	−	0.606434i	$-0.792599\pi$
$673$	−41.2551	−1.59027	−0.795133	+	0.606434i	$0.792599\pi$
$674$	0	0
$675$	−45.3059	−1.74383
$676$	0	0
$677$	7.57086	0.290972	0.145486	−	0.989360i	$-0.453525\pi$
$677$	7.57086	0.290972	0.145486	+	0.989360i	$0.453525\pi$
$678$	0	0
$679$	0.00507406	0.000194724 0
$680$	0	0
$681$	21.6709	0.830430
$682$	0	0
$683$	−32.3734	−1.23873	−0.619367	−	0.785101i	$-0.712611\pi$
$683$	−32.3734	−1.23873	−0.619367	+	0.785101i	$0.712611\pi$
$684$	0	0
$685$	−44.0261	−1.68215
$686$	0	0
$687$	−43.9662	−1.67742
$688$	0	0
$689$	1.19056	0.0453566
$690$	0	0
$691$	14.2649	0.542661	0.271331	−	0.962486i	$-0.412536\pi$
$691$	14.2649	0.542661	0.271331	+	0.962486i	$0.412536\pi$
$692$	0	0
$693$	0.130705	0.00496508
$694$	0	0
$695$	50.3855	1.91123
$696$	0	0
$697$	−2.08683	−0.0790443
$698$	0	0
$699$	30.1450	1.14019
$700$	0	0
$701$	−25.9907	−0.981656	−0.490828	−	0.871256i	$-0.663306\pi$
$701$	−25.9907	−0.981656	−0.490828	+	0.871256i	$0.663306\pi$
$702$	0	0
$703$	−8.98475	−0.338866
$704$	0	0
$705$	−70.8553	−2.66856
$706$	0	0
$707$	0.761363	0.0286340
$708$	0	0
$709$	39.7232	1.49184	0.745919	−	0.666037i	$-0.232011\pi$
$709$	39.7232	1.49184	0.745919	+	0.666037i	$0.232011\pi$
$710$	0	0
$711$	1.57928	0.0592275
$712$	0	0
$713$	31.0500	1.16283
$714$	0	0
$715$	6.17342	0.230873
$716$	0	0
$717$	49.3565	1.84325
$718$	0	0
$719$	33.6497	1.25492	0.627462	−	0.778647i	$-0.284094\pi$
$719$	33.6497	1.25492	0.627462	+	0.778647i	$0.284094\pi$
$720$	0	0
$721$	−7.84837	−0.292289
$722$	0	0
$723$	4.64628	0.172797
$724$	0	0
$725$	44.9704	1.67016
$726$	0	0
$727$	38.5806	1.43087	0.715437	−	0.698677i	$-0.246228\pi$
$727$	38.5806	1.43087	0.715437	+	0.698677i	$0.246228\pi$
$728$	0	0
$729$	28.2101	1.04482
$730$	0	0
$731$	2.97022	0.109857
$732$	0	0
$733$	10.7138	0.395723	0.197861	−	0.980230i	$-0.436600\pi$
$733$	10.7138	0.395723	0.197861	+	0.980230i	$0.436600\pi$
$734$	0	0
$735$	−41.0767	−1.51514
$736$	0	0
$737$	−7.55316	−0.278224
$738$	0	0
$739$	−20.6355	−0.759090	−0.379545	−	0.925173i	$-0.623919\pi$
$739$	−20.6355	−0.759090	−0.379545	+	0.925173i	$0.623919\pi$
$740$	0	0
$741$	2.00997	0.0738380
$742$	0	0
$743$	3.90242	0.143166	0.0715829	−	0.997435i	$-0.477195\pi$
$743$	3.90242	0.143166	0.0715829	+	0.997435i	$0.477195\pi$
$744$	0	0
$745$	−83.5760	−3.06199
$746$	0	0
$747$	−2.23601	−0.0818111
$748$	0	0
$749$	2.57177	0.0939706
$750$	0	0
$751$	1.68669	0.0615480	0.0307740	−	0.999526i	$-0.490203\pi$
$751$	1.68669	0.0615480	0.0307740	+	0.999526i	$0.490203\pi$
$752$	0	0
$753$	−45.5317	−1.65927
$754$	0	0
$755$	−73.3562	−2.66970
$756$	0	0
$757$	3.45222	0.125473	0.0627366	−	0.998030i	$-0.480017\pi$
$757$	3.45222	0.125473	0.0627366	+	0.998030i	$0.480017\pi$
$758$	0	0
$759$	12.8660	0.467007
$760$	0	0
$761$	−0.221484	−0.00802878	−0.00401439	−	0.999992i	$-0.501278\pi$
$761$	−0.221484	−0.00802878	−0.00401439	+	0.999992i	$0.501278\pi$
$762$	0	0
$763$	−1.99911	−0.0723728
$764$	0	0
$765$	−0.763432	−0.0276019
$766$	0	0
$767$	12.5695	0.453859
$768$	0	0
$769$	−45.0965	−1.62622	−0.813110	−	0.582110i	$-0.802227\pi$
$769$	−45.0965	−1.62622	−0.813110	+	0.582110i	$0.802227\pi$
$770$	0	0
$771$	10.1347	0.364993
$772$	0	0
$773$	1.41962	0.0510604	0.0255302	−	0.999674i	$-0.491873\pi$
$773$	1.41962	0.0510604	0.0255302	+	0.999674i	$0.491873\pi$
$774$	0	0
$775$	48.9531	1.75845
$776$	0	0
$777$	9.38575	0.336712
$778$	0	0
$779$	−1.50551	−0.0539404
$780$	0	0
$781$	4.97681	0.178084
$782$	0	0
$783$	−28.0680	−1.00307
$784$	0	0
$785$	29.5821	1.05583
$786$	0	0
$787$	21.3205	0.759994	0.379997	−	0.924988i	$-0.375925\pi$
$787$	21.3205	0.759994	0.379997	+	0.924988i	$0.375925\pi$
$788$	0	0
$789$	−6.67168	−0.237518
$790$	0	0
$791$	−6.94026	−0.246767
$792$	0	0
$793$	1.61491	0.0573471
$794$	0	0
$795$	6.20763	0.220162
$796$	0	0
$797$	13.9415	0.493832	0.246916	−	0.969037i	$-0.420583\pi$
$797$	13.9415	0.493832	0.246916	+	0.969037i	$0.420583\pi$
$798$	0	0
$799$	−15.8216	−0.559728
$800$	0	0
$801$	−0.410874	−0.0145175
$802$	0	0
$803$	5.91517	0.208742
$804$	0	0
$805$	−12.2950	−0.433342
$806$	0	0
$807$	−33.9884	−1.19645
$808$	0	0
$809$	−18.6021	−0.654016	−0.327008	−	0.945022i	$-0.606040\pi$
$809$	−18.6021	−0.654016	−0.327008	+	0.945022i	$0.606040\pi$
$810$	0	0
$811$	17.6690	0.620442	0.310221	−	0.950664i	$-0.399597\pi$
$811$	17.6690	0.620442	0.310221	+	0.950664i	$0.399597\pi$
$812$	0	0
$813$	−27.4300	−0.962011
$814$	0	0
$815$	22.5489	0.789854
$816$	0	0
$817$	2.14281	0.0749676
$818$	0	0
$819$	0.110345	0.00385578
$820$	0	0
$821$	−9.01292	−0.314553	−0.157277	−	0.987555i	$-0.550271\pi$
$821$	−9.01292	−0.314553	−0.157277	+	0.987555i	$0.550271\pi$
$822$	0	0
$823$	17.1693	0.598483	0.299242	−	0.954177i	$-0.403266\pi$
$823$	17.1693	0.598483	0.299242	+	0.954177i	$0.403266\pi$
$824$	0	0
$825$	20.2844	0.706213
$826$	0	0
$827$	53.6519	1.86566	0.932830	−	0.360316i	$-0.117331\pi$
$827$	53.6519	1.86566	0.932830	+	0.360316i	$0.117331\pi$
$828$	0	0
$829$	49.5707	1.72166	0.860830	−	0.508892i	$-0.169945\pi$
$829$	49.5707	1.72166	0.860830	+	0.508892i	$0.169945\pi$
$830$	0	0
$831$	−31.4295	−1.09028
$832$	0	0
$833$	−9.17220	−0.317798
$834$	0	0
$835$	35.1363	1.21594
$836$	0	0
$837$	−30.5538	−1.05609
$838$	0	0
$839$	−12.4878	−0.431127	−0.215563	−	0.976490i	$-0.569159\pi$
$839$	−12.4878	−0.431127	−0.215563	+	0.976490i	$0.569159\pi$
$840$	0	0
$841$	−1.13989	−0.0393066
$842$	0	0
$843$	−12.1552	−0.418648
$844$	0	0
$845$	−42.5885	−1.46509
$846$	0	0
$847$	5.57584	0.191588
$848$	0	0
$849$	−43.4294	−1.49049
$850$	0	0
$851$	−48.5538	−1.66440
$852$	0	0
$853$	−16.8636	−0.577398	−0.288699	−	0.957420i	$-0.593223\pi$
$853$	−16.8636	−0.577398	−0.288699	+	0.957420i	$0.593223\pi$
$854$	0	0
$855$	−0.550765	−0.0188358
$856$	0	0
$857$	−24.7530	−0.845547	−0.422773	−	0.906235i	$-0.638943\pi$
$857$	−24.7530	−0.845547	−0.422773	+	0.906235i	$0.638943\pi$
$858$	0	0
$859$	36.6424	1.25022	0.625111	−	0.780536i	$-0.285054\pi$
$859$	36.6424	1.25022	0.625111	+	0.780536i	$0.285054\pi$
$860$	0	0
$861$	1.57270	0.0535975
$862$	0	0
$863$	−25.4626	−0.866756	−0.433378	−	0.901212i	$-0.642678\pi$
$863$	−25.4626	−0.866756	−0.433378	+	0.901212i	$0.642678\pi$
$864$	0	0
$865$	−22.6253	−0.769283
$866$	0	0
$867$	−25.4566	−0.864552
$868$	0	0
$869$	−14.8685	−0.504380
$870$	0	0
$871$	−6.37662	−0.216063
$872$	0	0
$873$	0.00122832	4.15722e−5 0
$874$	0	0
$875$	−8.00836	−0.270732
$876$	0	0
$877$	−14.1586	−0.478101	−0.239051	−	0.971007i	$-0.576836\pi$
$877$	−14.1586	−0.478101	−0.239051	+	0.971007i	$0.576836\pi$
$878$	0	0
$879$	−31.1650	−1.05117
$880$	0	0
$881$	−51.6551	−1.74031	−0.870153	−	0.492782i	$-0.835980\pi$
$881$	−51.6551	−1.74031	−0.870153	+	0.492782i	$0.835980\pi$
$882$	0	0
$883$	−28.3086	−0.952659	−0.476330	−	0.879267i	$-0.658033\pi$
$883$	−28.3086	−0.952659	−0.476330	+	0.879267i	$0.658033\pi$
$884$	0	0
$885$	65.5381	2.20304
$886$	0	0
$887$	−3.74032	−0.125588	−0.0627939	−	0.998027i	$-0.520001\pi$
$887$	−3.74032	−0.125588	−0.0627939	+	0.998027i	$0.520001\pi$
$888$	0	0
$889$	−0.440731	−0.0147816
$890$	0	0
$891$	−12.0267	−0.402909
$892$	0	0
$893$	−11.4142	−0.381963
$894$	0	0
$895$	26.6416	0.890531
$896$	0	0
$897$	10.8619	0.362668
$898$	0	0
$899$	30.3275	1.01148
$900$	0	0
$901$	1.38613	0.0461787
$902$	0	0
$903$	−2.23845	−0.0744910
$904$	0	0
$905$	56.8215	1.88881
$906$	0	0
$907$	16.0865	0.534143	0.267071	−	0.963677i	$-0.413944\pi$
$907$	16.0865	0.534143	0.267071	+	0.963677i	$0.413944\pi$
$908$	0	0
$909$	0.184309	0.00611315
$910$	0	0
$911$	10.9948	0.364276	0.182138	−	0.983273i	$-0.441698\pi$
$911$	10.9948	0.364276	0.182138	+	0.983273i	$0.441698\pi$
$912$	0	0
$913$	21.0514	0.696701
$914$	0	0
$915$	8.42021	0.278364
$916$	0	0
$917$	3.83291	0.126574
$918$	0	0
$919$	−13.2787	−0.438023	−0.219012	−	0.975722i	$-0.570283\pi$
$919$	−13.2787	−0.438023	−0.219012	+	0.975722i	$0.570283\pi$
$920$	0	0
$921$	−45.2033	−1.48950
$922$	0	0
$923$	4.20158	0.138297
$924$	0	0
$925$	−76.5493	−2.51693
$926$	0	0
$927$	−1.89992	−0.0624015
$928$	0	0
$929$	−6.58027	−0.215891	−0.107946	−	0.994157i	$-0.534427\pi$
$929$	−6.58027	−0.215891	−0.107946	+	0.994157i	$0.534427\pi$
$930$	0	0
$931$	−6.61713	−0.216868
$932$	0	0
$933$	−18.2785	−0.598413
$934$	0	0
$935$	7.18753	0.235057
$936$	0	0
$937$	−35.8070	−1.16976	−0.584882	−	0.811118i	$-0.698859\pi$
$937$	−35.8070	−1.16976	−0.584882	+	0.811118i	$0.698859\pi$
$938$	0	0
$939$	−26.1727	−0.854115
$940$	0	0
$941$	−56.6114	−1.84548	−0.922740	−	0.385424i	$-0.874055\pi$
$941$	−56.6114	−1.84548	−0.922740	+	0.385424i	$0.874055\pi$
$942$	0	0
$943$	−8.13580	−0.264938
$944$	0	0
$945$	12.0985	0.393564
$946$	0	0
$947$	15.3634	0.499244	0.249622	−	0.968343i	$-0.419694\pi$
$947$	15.3634	0.499244	0.249622	+	0.968343i	$0.419694\pi$
$948$	0	0
$949$	4.99377	0.162105
$950$	0	0
$951$	14.3983	0.466896
$952$	0	0
$953$	46.1979	1.49650	0.748249	−	0.663418i	$-0.230895\pi$
$953$	46.1979	1.49650	0.748249	+	0.663418i	$0.230895\pi$
$954$	0	0
$955$	85.5472	2.76824
$956$	0	0
$957$	12.5666	0.406221
$958$	0	0
$959$	7.40880	0.239243
$960$	0	0
$961$	2.01342	0.0649491
$962$	0	0
$963$	0.622570	0.0200620
$964$	0	0
$965$	−44.9681	−1.44757
$966$	0	0
$967$	34.5248	1.11024	0.555122	−	0.831769i	$-0.312672\pi$
$967$	34.5248	1.11024	0.555122	+	0.831769i	$0.312672\pi$
$968$	0	0
$969$	2.34014	0.0751762
$970$	0	0
$971$	0.710877	0.0228131	0.0114066	−	0.999935i	$-0.496369\pi$
$971$	0.710877	0.0228131	0.0114066	+	0.999935i	$0.496369\pi$
$972$	0	0
$973$	−8.47897	−0.271823
$974$	0	0
$975$	17.1247	0.548431
$976$	0	0
$977$	5.85594	0.187348	0.0936741	−	0.995603i	$-0.470139\pi$
$977$	5.85594	0.187348	0.0936741	+	0.995603i	$0.470139\pi$
$978$	0	0
$979$	3.86828	0.123631
$980$	0	0
$981$	−0.483941	−0.0154511
$982$	0	0
$983$	20.7070	0.660450	0.330225	−	0.943902i	$-0.392875\pi$
$983$	20.7070	0.660450	0.330225	+	0.943902i	$0.392875\pi$
$984$	0	0
$985$	56.8076	1.81004
$986$	0	0
$987$	11.9237	0.379534
$988$	0	0
$989$	11.5798	0.368216
$990$	0	0
$991$	−8.61542	−0.273678	−0.136839	−	0.990593i	$-0.543694\pi$
$991$	−8.61542	−0.273678	−0.136839	+	0.990593i	$0.543694\pi$
$992$	0	0
$993$	56.0182	1.77768
$994$	0	0
$995$	−78.4057	−2.48563
$996$	0	0
$997$	26.4868	0.838845	0.419422	−	0.907791i	$-0.362233\pi$
$997$	26.4868	0.838845	0.419422	+	0.907791i	$0.362233\pi$
$998$	0	0
$999$	47.7778	1.51162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.e.1.14	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.14	✓	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)

\(f(q)\)	\(=\)	\(q+1.68826 q^{3} +3.67694 q^{5} -0.618763 q^{7} -0.149789 q^{9} +O(q^{10})\)	\(q+1.68826 q^{3} +3.67694 q^{5} -0.618763 q^{7} -0.149789 q^{9} +1.41023 q^{11} +1.19056 q^{13} +6.20763 q^{15} +1.38613 q^{17} +1.00000 q^{19} -1.04463 q^{21} +5.40402 q^{23} +8.51991 q^{25} -5.31765 q^{27} +5.27827 q^{29} +5.74573 q^{31} +2.38082 q^{33} -2.27516 q^{35} -8.98475 q^{37} +2.00997 q^{39} -1.50551 q^{41} +2.14281 q^{43} -0.550765 q^{45} -11.4142 q^{47} -6.61713 q^{49} +2.34014 q^{51} +1.00000 q^{53} +5.18532 q^{55} +1.68826 q^{57} +10.5577 q^{59} +1.35643 q^{61} +0.0926839 q^{63} +4.37761 q^{65} -5.35599 q^{67} +9.12337 q^{69} +3.52908 q^{71} +4.19448 q^{73} +14.3838 q^{75} -0.872596 q^{77} -10.5434 q^{79} -8.52820 q^{81} +14.9277 q^{83} +5.09672 q^{85} +8.91107 q^{87} +2.74302 q^{89} -0.736673 q^{91} +9.70027 q^{93} +3.67694 q^{95} -0.00820032 q^{97} -0.211236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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