LMFDB - Embedded newform 4028.2.a.d.1.1

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:	$1$
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} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
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			1.1
Root			$3.25955$ 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)

$f(q)$	$=$	$q-3.25955 q^{3} +3.01882 q^{5} +1.05463 q^{7} +7.62467 q^{9} +O(q^{10})$	$q-3.25955 q^{3} +3.01882 q^{5} +1.05463 q^{7} +7.62467 q^{9} +0.331559 q^{11} +5.94995 q^{13} -9.83998 q^{15} -5.88242 q^{17} -1.00000 q^{19} -3.43761 q^{21} -7.02557 q^{23} +4.11325 q^{25} -15.0743 q^{27} -0.630623 q^{29} +1.75086 q^{31} -1.08073 q^{33} +3.18372 q^{35} -8.41283 q^{37} -19.3942 q^{39} -3.43168 q^{41} -12.5011 q^{43} +23.0175 q^{45} -11.6939 q^{47} -5.88776 q^{49} +19.1741 q^{51} +1.00000 q^{53} +1.00092 q^{55} +3.25955 q^{57} +5.04111 q^{59} +8.88242 q^{61} +8.04118 q^{63} +17.9618 q^{65} -9.48288 q^{67} +22.9002 q^{69} +3.45336 q^{71} -11.7327 q^{73} -13.4073 q^{75} +0.349671 q^{77} +1.29391 q^{79} +26.2616 q^{81} +14.1005 q^{83} -17.7579 q^{85} +2.05555 q^{87} +0.759900 q^{89} +6.27497 q^{91} -5.70701 q^{93} -3.01882 q^{95} -7.99724 q^{97} +2.52803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + 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
$3$	−3.25955	−1.88190	−0.940951	−	0.338542i	$-0.890066\pi$
$3$	−3.25955	−1.88190	−0.940951	+	0.338542i	$0.890066\pi$
$4$	0	0
$5$	3.01882	1.35006	0.675028	−	0.737792i	$-0.264132\pi$
$5$	3.01882	1.35006	0.675028	+	0.737792i	$0.264132\pi$
$6$	0	0
$7$	1.05463	0.398611	0.199306	−	0.979937i	$-0.436131\pi$
$7$	1.05463	0.398611	0.199306	+	0.979937i	$0.436131\pi$
$8$	0	0
$9$	7.62467	2.54156
$10$	0	0
$11$	0.331559	0.0999688	0.0499844	−	0.998750i	$-0.484083\pi$
$11$	0.331559	0.0999688	0.0499844	+	0.998750i	$0.484083\pi$
$12$	0	0
$13$	5.94995	1.65022	0.825110	−	0.564972i	$-0.191113\pi$
$13$	5.94995	1.65022	0.825110	+	0.564972i	$0.191113\pi$
$14$	0	0
$15$	−9.83998	−2.54067
$16$	0	0
$17$	−5.88242	−1.42670	−0.713349	−	0.700809i	$-0.752822\pi$
$17$	−5.88242	−1.42670	−0.713349	+	0.700809i	$0.752822\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−3.43761	−0.750147
$22$	0	0
$23$	−7.02557	−1.46493	−0.732466	−	0.680804i	$-0.761631\pi$
$23$	−7.02557	−1.46493	−0.732466	+	0.680804i	$0.761631\pi$
$24$	0	0
$25$	4.11325	0.822649
$26$	0	0
$27$	−15.0743	−2.90106
$28$	0	0
$29$	−0.630623	−0.117104	−0.0585518	−	0.998284i	$-0.518648\pi$
$29$	−0.630623	−0.117104	−0.0585518	+	0.998284i	$0.518648\pi$
$30$	0	0
$31$	1.75086	0.314463	0.157232	−	0.987562i	$-0.449743\pi$
$31$	1.75086	0.314463	0.157232	+	0.987562i	$0.449743\pi$
$32$	0	0
$33$	−1.08073	−0.188132
$34$	0	0
$35$	3.18372	0.538147
$36$	0	0
$37$	−8.41283	−1.38306	−0.691531	−	0.722347i	$-0.743063\pi$
$37$	−8.41283	−1.38306	−0.691531	+	0.722347i	$0.743063\pi$
$38$	0	0
$39$	−19.3942	−3.10555
$40$	0	0
$41$	−3.43168	−0.535938	−0.267969	−	0.963427i	$-0.586353\pi$
$41$	−3.43168	−0.535938	−0.267969	+	0.963427i	$0.586353\pi$
$42$	0	0
$43$	−12.5011	−1.90641	−0.953203	−	0.302333i	$-0.902235\pi$
$43$	−12.5011	−1.90641	−0.953203	+	0.302333i	$0.902235\pi$
$44$	0	0
$45$	23.0175	3.43124
$46$	0	0
$47$	−11.6939	−1.70573	−0.852866	−	0.522130i	$-0.825137\pi$
$47$	−11.6939	−1.70573	−0.852866	+	0.522130i	$0.825137\pi$
$48$	0	0
$49$	−5.88776	−0.841109
$50$	0	0
$51$	19.1741	2.68490
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	1.00092	0.134963
$56$	0	0
$57$	3.25955	0.431738
$58$	0	0
$59$	5.04111	0.656296	0.328148	−	0.944626i	$-0.393576\pi$
$59$	5.04111	0.656296	0.328148	+	0.944626i	$0.393576\pi$
$60$	0	0
$61$	8.88242	1.13728	0.568639	−	0.822587i	$-0.307470\pi$
$61$	8.88242	1.13728	0.568639	+	0.822587i	$0.307470\pi$
$62$	0	0
$63$	8.04118	1.01309
$64$	0	0
$65$	17.9618	2.22789
$66$	0	0
$67$	−9.48288	−1.15852	−0.579259	−	0.815144i	$-0.696658\pi$
$67$	−9.48288	−1.15852	−0.579259	+	0.815144i	$0.696658\pi$
$68$	0	0
$69$	22.9002	2.75686
$70$	0	0
$71$	3.45336	0.409838	0.204919	−	0.978779i	$-0.434307\pi$
$71$	3.45336	0.409838	0.204919	+	0.978779i	$0.434307\pi$
$72$	0	0
$73$	−11.7327	−1.37321	−0.686604	−	0.727032i	$-0.740899\pi$
$73$	−11.7327	−1.37321	−0.686604	+	0.727032i	$0.740899\pi$
$74$	0	0
$75$	−13.4073	−1.54815
$76$	0	0
$77$	0.349671	0.0398487
$78$	0	0
$79$	1.29391	0.145576	0.0727882	−	0.997347i	$-0.476810\pi$
$79$	1.29391	0.145576	0.0727882	+	0.997347i	$0.476810\pi$
$80$	0	0
$81$	26.2616	2.91795
$82$	0	0
$83$	14.1005	1.54773	0.773866	−	0.633349i	$-0.218320\pi$
$83$	14.1005	1.54773	0.773866	+	0.633349i	$0.218320\pi$
$84$	0	0
$85$	−17.7579	−1.92612
$86$	0	0
$87$	2.05555	0.220378
$88$	0	0
$89$	0.759900	0.0805492	0.0402746	−	0.999189i	$-0.487177\pi$
$89$	0.759900	0.0805492	0.0402746	+	0.999189i	$0.487177\pi$
$90$	0	0
$91$	6.27497	0.657796
$92$	0	0
$93$	−5.70701	−0.591789
$94$	0	0
$95$	−3.01882	−0.309724
$96$	0	0
$97$	−7.99724	−0.811996	−0.405998	−	0.913874i	$-0.633076\pi$
$97$	−7.99724	−0.811996	−0.405998	+	0.913874i	$0.633076\pi$
$98$	0	0
$99$	2.52803	0.254076
$100$	0	0
$101$	−8.67774	−0.863467	−0.431734	−	0.902001i	$-0.642098\pi$
$101$	−8.67774	−0.863467	−0.431734	+	0.902001i	$0.642098\pi$
$102$	0	0
$103$	−19.5852	−1.92978	−0.964891	−	0.262650i	$-0.915403\pi$
$103$	−19.5852	−1.92978	−0.964891	+	0.262650i	$0.915403\pi$
$104$	0	0
$105$	−10.3775	−1.01274
$106$	0	0
$107$	17.7875	1.71958	0.859790	−	0.510648i	$-0.170595\pi$
$107$	17.7875	1.71958	0.859790	+	0.510648i	$0.170595\pi$
$108$	0	0
$109$	−6.71722	−0.643393	−0.321697	−	0.946843i	$-0.604253\pi$
$109$	−6.71722	−0.643393	−0.321697	+	0.946843i	$0.604253\pi$
$110$	0	0
$111$	27.4221	2.60279
$112$	0	0
$113$	13.9055	1.30812	0.654060	−	0.756443i	$-0.273064\pi$
$113$	13.9055	1.30812	0.654060	+	0.756443i	$0.273064\pi$
$114$	0	0
$115$	−21.2089	−1.97774
$116$	0	0
$117$	45.3664	4.19413
$118$	0	0
$119$	−6.20376	−0.568697
$120$	0	0
$121$	−10.8901	−0.990006
$122$	0	0
$123$	11.1857	1.00858
$124$	0	0
$125$	−2.67695	−0.239434
$126$	0	0
$127$	17.9613	1.59380	0.796902	−	0.604108i	$-0.206471\pi$
$127$	17.9613	1.59380	0.796902	+	0.604108i	$0.206471\pi$
$128$	0	0
$129$	40.7481	3.58767
$130$	0	0
$131$	−20.4071	−1.78297	−0.891487	−	0.453046i	$-0.850337\pi$
$131$	−20.4071	−1.78297	−0.891487	+	0.453046i	$0.850337\pi$
$132$	0	0
$133$	−1.05463	−0.0914477
$134$	0	0
$135$	−45.5067	−3.91659
$136$	0	0
$137$	20.2060	1.72631	0.863157	−	0.504936i	$-0.168484\pi$
$137$	20.2060	1.72631	0.863157	+	0.504936i	$0.168484\pi$
$138$	0	0
$139$	−15.3310	−1.30036	−0.650178	−	0.759782i	$-0.725306\pi$
$139$	−15.3310	−1.30036	−0.650178	+	0.759782i	$0.725306\pi$
$140$	0	0
$141$	38.1169	3.21002
$142$	0	0
$143$	1.97276	0.164970
$144$	0	0
$145$	−1.90373	−0.158096
$146$	0	0
$147$	19.1915	1.58289
$148$	0	0
$149$	−10.8655	−0.890136	−0.445068	−	0.895497i	$-0.646820\pi$
$149$	−10.8655	−0.890136	−0.445068	+	0.895497i	$0.646820\pi$
$150$	0	0
$151$	14.9050	1.21295	0.606477	−	0.795101i	$-0.292582\pi$
$151$	14.9050	1.21295	0.606477	+	0.795101i	$0.292582\pi$
$152$	0	0
$153$	−44.8515	−3.62603
$154$	0	0
$155$	5.28552	0.424543
$156$	0	0
$157$	−7.90326	−0.630749	−0.315374	−	0.948967i	$-0.602130\pi$
$157$	−7.90326	−0.630749	−0.315374	+	0.948967i	$0.602130\pi$
$158$	0	0
$159$	−3.25955	−0.258499
$160$	0	0
$161$	−7.40935	−0.583938
$162$	0	0
$163$	−9.78577	−0.766480	−0.383240	−	0.923649i	$-0.625192\pi$
$163$	−9.78577	−0.766480	−0.383240	+	0.923649i	$0.625192\pi$
$164$	0	0
$165$	−3.26253	−0.253988
$166$	0	0
$167$	4.07577	0.315393	0.157696	−	0.987488i	$-0.449593\pi$
$167$	4.07577	0.315393	0.157696	+	0.987488i	$0.449593\pi$
$168$	0	0
$169$	22.4019	1.72322
$170$	0	0
$171$	−7.62467	−0.583073
$172$	0	0
$173$	4.69793	0.357177	0.178588	−	0.983924i	$-0.442847\pi$
$173$	4.69793	0.357177	0.178588	+	0.983924i	$0.442847\pi$
$174$	0	0
$175$	4.33794	0.327917
$176$	0	0
$177$	−16.4317	−1.23509
$178$	0	0
$179$	−10.2545	−0.766454	−0.383227	−	0.923654i	$-0.625187\pi$
$179$	−10.2545	−0.766454	−0.383227	+	0.923654i	$0.625187\pi$
$180$	0	0
$181$	2.96755	0.220576	0.110288	−	0.993900i	$-0.464823\pi$
$181$	2.96755	0.220576	0.110288	+	0.993900i	$0.464823\pi$
$182$	0	0
$183$	−28.9527	−2.14024
$184$	0	0
$185$	−25.3968	−1.86721
$186$	0	0
$187$	−1.95037	−0.142625
$188$	0	0
$189$	−15.8978	−1.15639
$190$	0	0
$191$	−2.27081	−0.164310	−0.0821550	−	0.996620i	$-0.526180\pi$
$191$	−2.27081	−0.164310	−0.0821550	+	0.996620i	$0.526180\pi$
$192$	0	0
$193$	8.42938	0.606760	0.303380	−	0.952870i	$-0.401885\pi$
$193$	8.42938	0.606760	0.303380	+	0.952870i	$0.401885\pi$
$194$	0	0
$195$	−58.5474	−4.19267
$196$	0	0
$197$	6.00992	0.428189	0.214095	−	0.976813i	$-0.431320\pi$
$197$	6.00992	0.428189	0.214095	+	0.976813i	$0.431320\pi$
$198$	0	0
$199$	27.3567	1.93927	0.969633	−	0.244565i	$-0.0786452\pi$
$199$	27.3567	1.93927	0.969633	+	0.244565i	$0.0786452\pi$
$200$	0	0
$201$	30.9099	2.18022
$202$	0	0
$203$	−0.665071	−0.0466788
$204$	0	0
$205$	−10.3596	−0.723547
$206$	0	0
$207$	−53.5676	−3.72321
$208$	0	0
$209$	−0.331559	−0.0229344
$210$	0	0
$211$	4.78615	0.329492	0.164746	−	0.986336i	$-0.447320\pi$
$211$	4.78615	0.329492	0.164746	+	0.986336i	$0.447320\pi$
$212$	0	0
$213$	−11.2564	−0.771275
$214$	0	0
$215$	−37.7386	−2.57375
$216$	0	0
$217$	1.84650	0.125349
$218$	0	0
$219$	38.2433	2.58424
$220$	0	0
$221$	−35.0001	−2.35436
$222$	0	0
$223$	−15.0186	−1.00572	−0.502859	−	0.864369i	$-0.667718\pi$
$223$	−15.0186	−1.00572	−0.502859	+	0.864369i	$0.667718\pi$
$224$	0	0
$225$	31.3621	2.09081
$226$	0	0
$227$	0.347715	0.0230786	0.0115393	−	0.999933i	$-0.496327\pi$
$227$	0.347715	0.0230786	0.0115393	+	0.999933i	$0.496327\pi$
$228$	0	0
$229$	−7.34107	−0.485112	−0.242556	−	0.970137i	$-0.577986\pi$
$229$	−7.34107	−0.485112	−0.242556	+	0.970137i	$0.577986\pi$
$230$	0	0
$231$	−1.13977	−0.0749913
$232$	0	0
$233$	−16.8167	−1.10170	−0.550849	−	0.834605i	$-0.685696\pi$
$233$	−16.8167	−1.10170	−0.550849	+	0.834605i	$0.685696\pi$
$234$	0	0
$235$	−35.3017	−2.30283
$236$	0	0
$237$	−4.21757	−0.273961
$238$	0	0
$239$	−7.82872	−0.506398	−0.253199	−	0.967414i	$-0.581483\pi$
$239$	−7.82872	−0.506398	−0.253199	+	0.967414i	$0.581483\pi$
$240$	0	0
$241$	1.61594	0.104092	0.0520458	−	0.998645i	$-0.483426\pi$
$241$	1.61594	0.104092	0.0520458	+	0.998645i	$0.483426\pi$
$242$	0	0
$243$	−40.3779	−2.59025
$244$	0	0
$245$	−17.7741	−1.13554
$246$	0	0
$247$	−5.94995	−0.378586
$248$	0	0
$249$	−45.9613	−2.91268
$250$	0	0
$251$	18.9517	1.19622	0.598111	−	0.801413i	$-0.295918\pi$
$251$	18.9517	1.19622	0.598111	+	0.801413i	$0.295918\pi$
$252$	0	0
$253$	−2.32939	−0.146448
$254$	0	0
$255$	57.8829	3.62477
$256$	0	0
$257$	−7.49487	−0.467517	−0.233759	−	0.972295i	$-0.575103\pi$
$257$	−7.49487	−0.467517	−0.233759	+	0.972295i	$0.575103\pi$
$258$	0	0
$259$	−8.87239	−0.551304
$260$	0	0
$261$	−4.80829	−0.297626
$262$	0	0
$263$	6.47807	0.399455	0.199727	−	0.979851i	$-0.435994\pi$
$263$	6.47807	0.399455	0.199727	+	0.979851i	$0.435994\pi$
$264$	0	0
$265$	3.01882	0.185444
$266$	0	0
$267$	−2.47693	−0.151586
$268$	0	0
$269$	12.0852	0.736850	0.368425	−	0.929657i	$-0.379897\pi$
$269$	12.0852	0.736850	0.368425	+	0.929657i	$0.379897\pi$
$270$	0	0
$271$	−19.0918	−1.15974	−0.579872	−	0.814708i	$-0.696897\pi$
$271$	−19.0918	−1.15974	−0.579872	+	0.814708i	$0.696897\pi$
$272$	0	0
$273$	−20.4536	−1.23791
$274$	0	0
$275$	1.36378	0.0822393
$276$	0	0
$277$	−22.9238	−1.37736	−0.688680	−	0.725065i	$-0.741809\pi$
$277$	−22.9238	−1.37736	−0.688680	+	0.725065i	$0.741809\pi$
$278$	0	0
$279$	13.3497	0.799226
$280$	0	0
$281$	−15.4236	−0.920095	−0.460048	−	0.887894i	$-0.652168\pi$
$281$	−15.4236	−0.920095	−0.460048	+	0.887894i	$0.652168\pi$
$282$	0	0
$283$	1.62130	0.0963765	0.0481883	−	0.998838i	$-0.484655\pi$
$283$	1.62130	0.0963765	0.0481883	+	0.998838i	$0.484655\pi$
$284$	0	0
$285$	9.83998	0.582870
$286$	0	0
$287$	−3.61914	−0.213631
$288$	0	0
$289$	17.6029	1.03546
$290$	0	0
$291$	26.0674	1.52810
$292$	0	0
$293$	10.1597	0.593538	0.296769	−	0.954949i	$-0.404091\pi$
$293$	10.1597	0.593538	0.296769	+	0.954949i	$0.404091\pi$
$294$	0	0
$295$	15.2182	0.886036
$296$	0	0
$297$	−4.99804	−0.290015
$298$	0	0
$299$	−41.8018	−2.41746
$300$	0	0
$301$	−13.1840	−0.759914
$302$	0	0
$303$	28.2855	1.62496
$304$	0	0
$305$	26.8144	1.53539
$306$	0	0
$307$	−30.8456	−1.76045	−0.880227	−	0.474553i	$-0.842610\pi$
$307$	−30.8456	−1.76045	−0.880227	+	0.474553i	$0.842610\pi$
$308$	0	0
$309$	63.8388	3.63166
$310$	0	0
$311$	25.3115	1.43529	0.717643	−	0.696411i	$-0.245221\pi$
$311$	25.3115	1.43529	0.717643	+	0.696411i	$0.245221\pi$
$312$	0	0
$313$	−16.4433	−0.929428	−0.464714	−	0.885461i	$-0.653843\pi$
$313$	−16.4433	−0.929428	−0.464714	+	0.885461i	$0.653843\pi$
$314$	0	0
$315$	24.2748	1.36773
$316$	0	0
$317$	−22.4750	−1.26232	−0.631162	−	0.775651i	$-0.717422\pi$
$317$	−22.4750	−1.26232	−0.631162	+	0.775651i	$0.717422\pi$
$318$	0	0
$319$	−0.209089	−0.0117067
$320$	0	0
$321$	−57.9791	−3.23608
$322$	0	0
$323$	5.88242	0.327307
$324$	0	0
$325$	24.4736	1.35755
$326$	0	0
$327$	21.8951	1.21080
$328$	0	0
$329$	−12.3327	−0.679924
$330$	0	0
$331$	−26.3487	−1.44825	−0.724127	−	0.689667i	$-0.757757\pi$
$331$	−26.3487	−1.44825	−0.724127	+	0.689667i	$0.757757\pi$
$332$	0	0
$333$	−64.1451	−3.51513
$334$	0	0
$335$	−28.6271	−1.56406
$336$	0	0
$337$	27.0126	1.47147	0.735736	−	0.677268i	$-0.236836\pi$
$337$	27.0126	1.47147	0.735736	+	0.677268i	$0.236836\pi$
$338$	0	0
$339$	−45.3257	−2.46175
$340$	0	0
$341$	0.580513	0.0314365
$342$	0	0
$343$	−13.5918	−0.733887
$344$	0	0
$345$	69.1314	3.72191
$346$	0	0
$347$	9.44339	0.506947	0.253474	−	0.967342i	$-0.418427\pi$
$347$	9.44339	0.506947	0.253474	+	0.967342i	$0.418427\pi$
$348$	0	0
$349$	32.3767	1.73308	0.866542	−	0.499104i	$-0.166338\pi$
$349$	32.3767	1.73308	0.866542	+	0.499104i	$0.166338\pi$
$350$	0	0
$351$	−89.6916	−4.78738
$352$	0	0
$353$	21.4662	1.14253	0.571265	−	0.820765i	$-0.306453\pi$
$353$	21.4662	1.14253	0.571265	+	0.820765i	$0.306453\pi$
$354$	0	0
$355$	10.4250	0.553304
$356$	0	0
$357$	20.2215	1.07023
$358$	0	0
$359$	−8.48430	−0.447784	−0.223892	−	0.974614i	$-0.571876\pi$
$359$	−8.48430	−0.447784	−0.223892	+	0.974614i	$0.571876\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	35.4967	1.86310
$364$	0	0
$365$	−35.4188	−1.85391
$366$	0	0
$367$	2.51304	0.131179	0.0655897	−	0.997847i	$-0.479107\pi$
$367$	2.51304	0.131179	0.0655897	+	0.997847i	$0.479107\pi$
$368$	0	0
$369$	−26.1654	−1.36212
$370$	0	0
$371$	1.05463	0.0547535
$372$	0	0
$373$	−10.2941	−0.533010	−0.266505	−	0.963834i	$-0.585869\pi$
$373$	−10.2941	−0.533010	−0.266505	+	0.963834i	$0.585869\pi$
$374$	0	0
$375$	8.72565	0.450591
$376$	0	0
$377$	−3.75217	−0.193247
$378$	0	0
$379$	9.99086	0.513196	0.256598	−	0.966518i	$-0.417398\pi$
$379$	9.99086	0.513196	0.256598	+	0.966518i	$0.417398\pi$
$380$	0	0
$381$	−58.5457	−2.99938
$382$	0	0
$383$	26.6459	1.36154	0.680771	−	0.732496i	$-0.261645\pi$
$383$	26.6459	1.36154	0.680771	+	0.732496i	$0.261645\pi$
$384$	0	0
$385$	1.05559	0.0537979
$386$	0	0
$387$	−95.3170	−4.84524
$388$	0	0
$389$	−11.4540	−0.580740	−0.290370	−	0.956914i	$-0.593778\pi$
$389$	−11.4540	−0.580740	−0.290370	+	0.956914i	$0.593778\pi$
$390$	0	0
$391$	41.3274	2.09001
$392$	0	0
$393$	66.5179	3.35538
$394$	0	0
$395$	3.90608	0.196536
$396$	0	0
$397$	2.62351	0.131670	0.0658351	−	0.997831i	$-0.479029\pi$
$397$	2.62351	0.131670	0.0658351	+	0.997831i	$0.479029\pi$
$398$	0	0
$399$	3.43761	0.172096
$400$	0	0
$401$	17.0531	0.851591	0.425796	−	0.904819i	$-0.359994\pi$
$401$	17.0531	0.851591	0.425796	+	0.904819i	$0.359994\pi$
$402$	0	0
$403$	10.4175	0.518933
$404$	0	0
$405$	79.2789	3.93940
$406$	0	0
$407$	−2.78935	−0.138263
$408$	0	0
$409$	18.6613	0.922744	0.461372	−	0.887207i	$-0.347357\pi$
$409$	18.6613	0.922744	0.461372	+	0.887207i	$0.347357\pi$
$410$	0	0
$411$	−65.8624	−3.24875
$412$	0	0
$413$	5.31648	0.261607
$414$	0	0
$415$	42.5668	2.08952
$416$	0	0
$417$	49.9721	2.44714
$418$	0	0
$419$	3.26616	0.159563	0.0797813	−	0.996812i	$-0.474578\pi$
$419$	3.26616	0.159563	0.0797813	+	0.996812i	$0.474578\pi$
$420$	0	0
$421$	5.73520	0.279517	0.139758	−	0.990186i	$-0.455367\pi$
$421$	5.73520	0.279517	0.139758	+	0.990186i	$0.455367\pi$
$422$	0	0
$423$	−89.1622	−4.33521
$424$	0	0
$425$	−24.1958	−1.17367
$426$	0	0
$427$	9.36763	0.453331
$428$	0	0
$429$	−6.43031	−0.310458
$430$	0	0
$431$	−31.7553	−1.52960	−0.764801	−	0.644267i	$-0.777163\pi$
$431$	−31.7553	−1.52960	−0.764801	+	0.644267i	$0.777163\pi$
$432$	0	0
$433$	14.2356	0.684121	0.342060	−	0.939678i	$-0.388875\pi$
$433$	14.2356	0.684121	0.342060	+	0.939678i	$0.388875\pi$
$434$	0	0
$435$	6.20531	0.297522
$436$	0	0
$437$	7.02557	0.336078
$438$	0	0
$439$	5.29394	0.252666	0.126333	−	0.991988i	$-0.459679\pi$
$439$	5.29394	0.252666	0.126333	+	0.991988i	$0.459679\pi$
$440$	0	0
$441$	−44.8923	−2.13773
$442$	0	0
$443$	39.0521	1.85542	0.927711	−	0.373298i	$-0.121773\pi$
$443$	39.0521	1.85542	0.927711	+	0.373298i	$0.121773\pi$
$444$	0	0
$445$	2.29400	0.108746
$446$	0	0
$447$	35.4166	1.67515
$448$	0	0
$449$	−12.6599	−0.597456	−0.298728	−	0.954338i	$-0.596562\pi$
$449$	−12.6599	−0.597456	−0.298728	+	0.954338i	$0.596562\pi$
$450$	0	0
$451$	−1.13780	−0.0535771
$452$	0	0
$453$	−48.5837	−2.28266
$454$	0	0
$455$	18.9430	0.888061
$456$	0	0
$457$	−39.5236	−1.84884	−0.924419	−	0.381379i	$-0.875449\pi$
$457$	−39.5236	−1.84884	−0.924419	+	0.381379i	$0.875449\pi$
$458$	0	0
$459$	88.6737	4.13893
$460$	0	0
$461$	−1.81260	−0.0844213	−0.0422106	−	0.999109i	$-0.513440\pi$
$461$	−1.81260	−0.0844213	−0.0422106	+	0.999109i	$0.513440\pi$
$462$	0	0
$463$	40.0534	1.86144	0.930719	−	0.365736i	$-0.119183\pi$
$463$	40.0534	1.86144	0.930719	+	0.365736i	$0.119183\pi$
$464$	0	0
$465$	−17.2284	−0.798948
$466$	0	0
$467$	−6.25240	−0.289327	−0.144663	−	0.989481i	$-0.546210\pi$
$467$	−6.25240	−0.289327	−0.144663	+	0.989481i	$0.546210\pi$
$468$	0	0
$469$	−10.0009	−0.461798
$470$	0	0
$471$	25.7611	1.18701
$472$	0	0
$473$	−4.14486	−0.190581
$474$	0	0
$475$	−4.11325	−0.188729
$476$	0	0
$477$	7.62467	0.349110
$478$	0	0
$479$	7.77440	0.355221	0.177611	−	0.984101i	$-0.443163\pi$
$479$	7.77440	0.355221	0.177611	+	0.984101i	$0.443163\pi$
$480$	0	0
$481$	−50.0560	−2.28236
$482$	0	0
$483$	24.1511	1.09891
$484$	0	0
$485$	−24.1422	−1.09624
$486$	0	0
$487$	8.07103	0.365733	0.182867	−	0.983138i	$-0.441462\pi$
$487$	8.07103	0.365733	0.182867	+	0.983138i	$0.441462\pi$
$488$	0	0
$489$	31.8972	1.44244
$490$	0	0
$491$	28.5768	1.28965	0.644826	−	0.764329i	$-0.276930\pi$
$491$	28.5768	1.28965	0.644826	+	0.764329i	$0.276930\pi$
$492$	0	0
$493$	3.70959	0.167071
$494$	0	0
$495$	7.63165	0.343017
$496$	0	0
$497$	3.64200	0.163366
$498$	0	0
$499$	−18.5531	−0.830551	−0.415275	−	0.909696i	$-0.636315\pi$
$499$	−18.5531	−0.830551	−0.415275	+	0.909696i	$0.636315\pi$
$500$	0	0
$501$	−13.2852	−0.593538
$502$	0	0
$503$	34.4682	1.53686	0.768430	−	0.639934i	$-0.221038\pi$
$503$	34.4682	1.53686	0.768430	+	0.639934i	$0.221038\pi$
$504$	0	0
$505$	−26.1965	−1.16573
$506$	0	0
$507$	−73.0202	−3.24294
$508$	0	0
$509$	6.39583	0.283490	0.141745	−	0.989903i	$-0.454729\pi$
$509$	6.39583	0.283490	0.141745	+	0.989903i	$0.454729\pi$
$510$	0	0
$511$	−12.3736	−0.547376
$512$	0	0
$513$	15.0743	0.665549
$514$	0	0
$515$	−59.1240	−2.60531
$516$	0	0
$517$	−3.87722	−0.170520
$518$	0	0
$519$	−15.3131	−0.672172
$520$	0	0
$521$	30.1952	1.32288	0.661438	−	0.750000i	$-0.269947\pi$
$521$	30.1952	1.32288	0.661438	+	0.750000i	$0.269947\pi$
$522$	0	0
$523$	−16.4307	−0.718463	−0.359231	−	0.933249i	$-0.616961\pi$
$523$	−16.4307	−0.718463	−0.359231	+	0.933249i	$0.616961\pi$
$524$	0	0
$525$	−14.1397	−0.617108
$526$	0	0
$527$	−10.2993	−0.448644
$528$	0	0
$529$	26.3586	1.14603
$530$	0	0
$531$	38.4368	1.66801
$532$	0	0
$533$	−20.4183	−0.884416
$534$	0	0
$535$	53.6971	2.32153
$536$	0	0
$537$	33.4249	1.44239
$538$	0	0
$539$	−1.95214	−0.0840847
$540$	0	0
$541$	−0.619098	−0.0266171	−0.0133086	−	0.999911i	$-0.504236\pi$
$541$	−0.619098	−0.0266171	−0.0133086	+	0.999911i	$0.504236\pi$
$542$	0	0
$543$	−9.67287	−0.415103
$544$	0	0
$545$	−20.2781	−0.868617
$546$	0	0
$547$	29.1904	1.24809	0.624046	−	0.781388i	$-0.285488\pi$
$547$	29.1904	1.24809	0.624046	+	0.781388i	$0.285488\pi$
$548$	0	0
$549$	67.7255	2.89045
$550$	0	0
$551$	0.630623	0.0268654
$552$	0	0
$553$	1.36459	0.0580284
$554$	0	0
$555$	82.7821	3.51391
$556$	0	0
$557$	0.976219	0.0413637	0.0206819	−	0.999786i	$-0.493416\pi$
$557$	0.976219	0.0413637	0.0206819	+	0.999786i	$0.493416\pi$
$558$	0	0
$559$	−74.3811	−3.14599
$560$	0	0
$561$	6.35733	0.268407
$562$	0	0
$563$	−1.41097	−0.0594652	−0.0297326	−	0.999558i	$-0.509466\pi$
$563$	−1.41097	−0.0594652	−0.0297326	+	0.999558i	$0.509466\pi$
$564$	0	0
$565$	41.9782	1.76603
$566$	0	0
$567$	27.6962	1.16313
$568$	0	0
$569$	−28.5801	−1.19814	−0.599071	−	0.800696i	$-0.704463\pi$
$569$	−28.5801	−1.19814	−0.599071	+	0.800696i	$0.704463\pi$
$570$	0	0
$571$	4.38818	0.183640	0.0918198	−	0.995776i	$-0.470732\pi$
$571$	4.38818	0.183640	0.0918198	+	0.995776i	$0.470732\pi$
$572$	0	0
$573$	7.40182	0.309215
$574$	0	0
$575$	−28.8979	−1.20512
$576$	0	0
$577$	13.0500	0.543277	0.271638	−	0.962399i	$-0.412435\pi$
$577$	13.0500	0.543277	0.271638	+	0.962399i	$0.412435\pi$
$578$	0	0
$579$	−27.4760	−1.14186
$580$	0	0
$581$	14.8708	0.616943
$582$	0	0
$583$	0.331559	0.0137318
$584$	0	0
$585$	136.953	5.66230
$586$	0	0
$587$	−11.0863	−0.457581	−0.228791	−	0.973476i	$-0.573477\pi$
$587$	−11.0863	−0.457581	−0.228791	+	0.973476i	$0.573477\pi$
$588$	0	0
$589$	−1.75086	−0.0721428
$590$	0	0
$591$	−19.5896	−0.805810
$592$	0	0
$593$	−14.9559	−0.614164	−0.307082	−	0.951683i	$-0.599353\pi$
$593$	−14.9559	−0.614164	−0.307082	+	0.951683i	$0.599353\pi$
$594$	0	0
$595$	−18.7280	−0.767773
$596$	0	0
$597$	−89.1706	−3.64951
$598$	0	0
$599$	11.4107	0.466230	0.233115	−	0.972449i	$-0.425108\pi$
$599$	11.4107	0.466230	0.233115	+	0.972449i	$0.425108\pi$
$600$	0	0
$601$	−23.4013	−0.954558	−0.477279	−	0.878752i	$-0.658377\pi$
$601$	−23.4013	−0.954558	−0.477279	+	0.878752i	$0.658377\pi$
$602$	0	0
$603$	−72.3038	−2.94444
$604$	0	0
$605$	−32.8751	−1.33656
$606$	0	0
$607$	0.210265	0.00853441	0.00426721	−	0.999991i	$-0.498642\pi$
$607$	0.210265	0.00853441	0.00426721	+	0.999991i	$0.498642\pi$
$608$	0	0
$609$	2.16783	0.0878450
$610$	0	0
$611$	−69.5782	−2.81483
$612$	0	0
$613$	−12.6532	−0.511056	−0.255528	−	0.966802i	$-0.582249\pi$
$613$	−12.6532	−0.511056	−0.255528	+	0.966802i	$0.582249\pi$
$614$	0	0
$615$	33.7677	1.36164
$616$	0	0
$617$	−10.2821	−0.413942	−0.206971	−	0.978347i	$-0.566361\pi$
$617$	−10.2821	−0.413942	−0.206971	+	0.978347i	$0.566361\pi$
$618$	0	0
$619$	−4.31481	−0.173427	−0.0867134	−	0.996233i	$-0.527636\pi$
$619$	−4.31481	−0.173427	−0.0867134	+	0.996233i	$0.527636\pi$
$620$	0	0
$621$	105.906	4.24985
$622$	0	0
$623$	0.801410	0.0321078
$624$	0	0
$625$	−28.6474	−1.14590
$626$	0	0
$627$	1.08073	0.0431603
$628$	0	0
$629$	49.4879	1.97321
$630$	0	0
$631$	−16.5934	−0.660573	−0.330286	−	0.943881i	$-0.607145\pi$
$631$	−16.5934	−0.660573	−0.330286	+	0.943881i	$0.607145\pi$
$632$	0	0
$633$	−15.6007	−0.620072
$634$	0	0
$635$	54.2217	2.15172
$636$	0	0
$637$	−35.0319	−1.38801
$638$	0	0
$639$	26.3307	1.04163
$640$	0	0
$641$	−21.2596	−0.839702	−0.419851	−	0.907593i	$-0.637918\pi$
$641$	−21.2596	−0.839702	−0.419851	+	0.907593i	$0.637918\pi$
$642$	0	0
$643$	−35.0880	−1.38374	−0.691869	−	0.722023i	$-0.743212\pi$
$643$	−35.0880	−1.38374	−0.691869	+	0.722023i	$0.743212\pi$
$644$	0	0
$645$	123.011	4.84355
$646$	0	0
$647$	−32.7770	−1.28860	−0.644298	−	0.764774i	$-0.722850\pi$
$647$	−32.7770	−1.28860	−0.644298	+	0.764774i	$0.722850\pi$
$648$	0	0
$649$	1.67142	0.0656091
$650$	0	0
$651$	−6.01876	−0.235894
$652$	0	0
$653$	−3.89507	−0.152426	−0.0762130	−	0.997092i	$-0.524283\pi$
$653$	−3.89507	−0.152426	−0.0762130	+	0.997092i	$0.524283\pi$
$654$	0	0
$655$	−61.6052	−2.40711
$656$	0	0
$657$	−89.4579	−3.49008
$658$	0	0
$659$	−35.6021	−1.38686	−0.693430	−	0.720524i	$-0.743901\pi$
$659$	−35.6021	−1.38686	−0.693430	+	0.720524i	$0.743901\pi$
$660$	0	0
$661$	13.0640	0.508131	0.254065	−	0.967187i	$-0.418232\pi$
$661$	13.0640	0.508131	0.254065	+	0.967187i	$0.418232\pi$
$662$	0	0
$663$	114.085	4.43068
$664$	0	0
$665$	−3.18372	−0.123459
$666$	0	0
$667$	4.43048	0.171549
$668$	0	0
$669$	48.9538	1.89266
$670$	0	0
$671$	2.94505	0.113692
$672$	0	0
$673$	−49.3187	−1.90110	−0.950548	−	0.310578i	$-0.899477\pi$
$673$	−49.3187	−1.90110	−0.950548	+	0.310578i	$0.899477\pi$
$674$	0	0
$675$	−62.0045	−2.38655
$676$	0	0
$677$	14.2902	0.549217	0.274609	−	0.961556i	$-0.411452\pi$
$677$	14.2902	0.549217	0.274609	+	0.961556i	$0.411452\pi$
$678$	0	0
$679$	−8.43409	−0.323671
$680$	0	0
$681$	−1.13339	−0.0434317
$682$	0	0
$683$	−5.75039	−0.220033	−0.110016	−	0.993930i	$-0.535090\pi$
$683$	−5.75039	−0.220033	−0.110016	+	0.993930i	$0.535090\pi$
$684$	0	0
$685$	60.9981	2.33062
$686$	0	0
$687$	23.9286	0.912933
$688$	0	0
$689$	5.94995	0.226675
$690$	0	0
$691$	2.28460	0.0869104	0.0434552	−	0.999055i	$-0.486163\pi$
$691$	2.28460	0.0869104	0.0434552	+	0.999055i	$0.486163\pi$
$692$	0	0
$693$	2.66612	0.101278
$694$	0	0
$695$	−46.2814	−1.75555
$696$	0	0
$697$	20.1866	0.764622
$698$	0	0
$699$	54.8149	2.07329
$700$	0	0
$701$	19.0938	0.721161	0.360581	−	0.932728i	$-0.382579\pi$
$701$	19.0938	0.721161	0.360581	+	0.932728i	$0.382579\pi$
$702$	0	0
$703$	8.41283	0.317296
$704$	0	0
$705$	115.068	4.33370
$706$	0	0
$707$	−9.15177	−0.344188
$708$	0	0
$709$	49.6459	1.86449	0.932245	−	0.361829i	$-0.117848\pi$
$709$	49.6459	1.86449	0.932245	+	0.361829i	$0.117848\pi$
$710$	0	0
$711$	9.86565	0.369991
$712$	0	0
$713$	−12.3008	−0.460667
$714$	0	0
$715$	5.95540	0.222719
$716$	0	0
$717$	25.5181	0.952992
$718$	0	0
$719$	47.2805	1.76326	0.881632	−	0.471937i	$-0.156445\pi$
$719$	47.2805	1.76326	0.881632	+	0.471937i	$0.156445\pi$
$720$	0	0
$721$	−20.6550	−0.769233
$722$	0	0
$723$	−5.26723	−0.195890
$724$	0	0
$725$	−2.59391	−0.0963352
$726$	0	0
$727$	29.5633	1.09644	0.548222	−	0.836333i	$-0.315305\pi$
$727$	29.5633	1.09644	0.548222	+	0.836333i	$0.315305\pi$
$728$	0	0
$729$	52.8292	1.95664
$730$	0	0
$731$	73.5370	2.71986
$732$	0	0
$733$	−6.93156	−0.256023	−0.128012	−	0.991773i	$-0.540859\pi$
$733$	−6.93156	−0.256023	−0.128012	+	0.991773i	$0.540859\pi$
$734$	0	0
$735$	57.9355	2.13698
$736$	0	0
$737$	−3.14413	−0.115816
$738$	0	0
$739$	−5.43256	−0.199840	−0.0999199	−	0.994995i	$-0.531859\pi$
$739$	−5.43256	−0.199840	−0.0999199	+	0.994995i	$0.531859\pi$
$740$	0	0
$741$	19.3942	0.712463
$742$	0	0
$743$	13.6644	0.501297	0.250648	−	0.968078i	$-0.419356\pi$
$743$	13.6644	0.501297	0.250648	+	0.968078i	$0.419356\pi$
$744$	0	0
$745$	−32.8009	−1.20173
$746$	0	0
$747$	107.512	3.93365
$748$	0	0
$749$	18.7591	0.685444
$750$	0	0
$751$	27.3953	0.999669	0.499834	−	0.866121i	$-0.333394\pi$
$751$	27.3953	0.999669	0.499834	+	0.866121i	$0.333394\pi$
$752$	0	0
$753$	−61.7741	−2.25117
$754$	0	0
$755$	44.9955	1.63756
$756$	0	0
$757$	28.6977	1.04303	0.521517	−	0.853241i	$-0.325366\pi$
$757$	28.6977	1.04303	0.521517	+	0.853241i	$0.325366\pi$
$758$	0	0
$759$	7.59277	0.275600
$760$	0	0
$761$	23.6984	0.859067	0.429534	−	0.903051i	$-0.358678\pi$
$761$	23.6984	0.859067	0.429534	+	0.903051i	$0.358678\pi$
$762$	0	0
$763$	−7.08416	−0.256464
$764$	0	0
$765$	−135.399	−4.89534
$766$	0	0
$767$	29.9943	1.08303
$768$	0	0
$769$	27.4336	0.989281	0.494641	−	0.869098i	$-0.335300\pi$
$769$	27.4336	0.989281	0.494641	+	0.869098i	$0.335300\pi$
$770$	0	0
$771$	24.4299	0.879822
$772$	0	0
$773$	−9.82676	−0.353444	−0.176722	−	0.984261i	$-0.556549\pi$
$773$	−9.82676	−0.353444	−0.176722	+	0.984261i	$0.556549\pi$
$774$	0	0
$775$	7.20171	0.258693
$776$	0	0
$777$	28.9200	1.03750
$778$	0	0
$779$	3.43168	0.122953
$780$	0	0
$781$	1.14499	0.0409710
$782$	0	0
$783$	9.50622	0.339725
$784$	0	0
$785$	−23.8585	−0.851546
$786$	0	0
$787$	−31.8143	−1.13406	−0.567029	−	0.823698i	$-0.691907\pi$
$787$	−31.8143	−1.13406	−0.567029	+	0.823698i	$0.691907\pi$
$788$	0	0
$789$	−21.1156	−0.751735
$790$	0	0
$791$	14.6651	0.521431
$792$	0	0
$793$	52.8500	1.87676
$794$	0	0
$795$	−9.83998	−0.348988
$796$	0	0
$797$	−3.73650	−0.132354	−0.0661768	−	0.997808i	$-0.521080\pi$
$797$	−3.73650	−0.132354	−0.0661768	+	0.997808i	$0.521080\pi$
$798$	0	0
$799$	68.7885	2.43356
$800$	0	0
$801$	5.79399	0.204720
$802$	0	0
$803$	−3.89008	−0.137278
$804$	0	0
$805$	−22.3674	−0.788349
$806$	0	0
$807$	−39.3925	−1.38668
$808$	0	0
$809$	27.3092	0.960141	0.480071	−	0.877230i	$-0.340611\pi$
$809$	27.3092	0.960141	0.480071	+	0.877230i	$0.340611\pi$
$810$	0	0
$811$	−23.7177	−0.832841	−0.416421	−	0.909172i	$-0.636716\pi$
$811$	−23.7177	−0.832841	−0.416421	+	0.909172i	$0.636716\pi$
$812$	0	0
$813$	62.2306	2.18252
$814$	0	0
$815$	−29.5414	−1.03479
$816$	0	0
$817$	12.5011	0.437359
$818$	0	0
$819$	47.8446	1.67183
$820$	0	0
$821$	−0.421489	−0.0147101	−0.00735504	−	0.999973i	$-0.502341\pi$
$821$	−0.421489	−0.0147101	−0.00735504	+	0.999973i	$0.502341\pi$
$822$	0	0
$823$	12.1082	0.422066	0.211033	−	0.977479i	$-0.432317\pi$
$823$	12.1082	0.422066	0.211033	+	0.977479i	$0.432317\pi$
$824$	0	0
$825$	−4.44532	−0.154766
$826$	0	0
$827$	−41.2512	−1.43444	−0.717222	−	0.696845i	$-0.754586\pi$
$827$	−41.2512	−1.43444	−0.717222	+	0.696845i	$0.754586\pi$
$828$	0	0
$829$	−24.7971	−0.861239	−0.430619	−	0.902534i	$-0.641705\pi$
$829$	−24.7971	−0.861239	−0.430619	+	0.902534i	$0.641705\pi$
$830$	0	0
$831$	74.7214	2.59206
$832$	0	0
$833$	34.6343	1.20001
$834$	0	0
$835$	12.3040	0.425798
$836$	0	0
$837$	−26.3930	−0.912277
$838$	0	0
$839$	−16.6884	−0.576148	−0.288074	−	0.957608i	$-0.593015\pi$
$839$	−16.6884	−0.576148	−0.288074	+	0.957608i	$0.593015\pi$
$840$	0	0
$841$	−28.6023	−0.986287
$842$	0	0
$843$	50.2740	1.73153
$844$	0	0
$845$	67.6272	2.32645
$846$	0	0
$847$	−11.4849	−0.394628
$848$	0	0
$849$	−5.28472	−0.181371
$850$	0	0
$851$	59.1049	2.02609
$852$	0	0
$853$	−35.6352	−1.22013	−0.610063	−	0.792353i	$-0.708856\pi$
$853$	−35.6352	−1.22013	−0.610063	+	0.792353i	$0.708856\pi$
$854$	0	0
$855$	−23.0175	−0.787181
$856$	0	0
$857$	14.4335	0.493039	0.246520	−	0.969138i	$-0.420713\pi$
$857$	14.4335	0.493039	0.246520	+	0.969138i	$0.420713\pi$
$858$	0	0
$859$	−20.9505	−0.714823	−0.357412	−	0.933947i	$-0.616341\pi$
$859$	−20.9505	−0.714823	−0.357412	+	0.933947i	$0.616341\pi$
$860$	0	0
$861$	11.7968	0.402033
$862$	0	0
$863$	−12.6525	−0.430697	−0.215349	−	0.976537i	$-0.569089\pi$
$863$	−12.6525	−0.430697	−0.215349	+	0.976537i	$0.569089\pi$
$864$	0	0
$865$	14.1822	0.482208
$866$	0	0
$867$	−57.3776	−1.94864
$868$	0	0
$869$	0.429008	0.0145531
$870$	0	0
$871$	−56.4227	−1.91181
$872$	0	0
$873$	−60.9763	−2.06373
$874$	0	0
$875$	−2.82318	−0.0954409
$876$	0	0
$877$	0.134343	0.00453646	0.00226823	−	0.999997i	$-0.499278\pi$
$877$	0.134343	0.00453646	0.00226823	+	0.999997i	$0.499278\pi$
$878$	0	0
$879$	−33.1162	−1.11698
$880$	0	0
$881$	28.1276	0.947644	0.473822	−	0.880621i	$-0.342874\pi$
$881$	28.1276	0.947644	0.473822	+	0.880621i	$0.342874\pi$
$882$	0	0
$883$	10.4037	0.350113	0.175057	−	0.984558i	$-0.443989\pi$
$883$	10.4037	0.350113	0.175057	+	0.984558i	$0.443989\pi$
$884$	0	0
$885$	−49.6044	−1.66743
$886$	0	0
$887$	33.0479	1.10964	0.554819	−	0.831971i	$-0.312787\pi$
$887$	33.0479	1.10964	0.554819	+	0.831971i	$0.312787\pi$
$888$	0	0
$889$	18.9424	0.635308
$890$	0	0
$891$	8.70727	0.291704
$892$	0	0
$893$	11.6939	0.391322
$894$	0	0
$895$	−30.9563	−1.03476
$896$	0	0
$897$	136.255	4.54942
$898$	0	0
$899$	−1.10413	−0.0368248
$900$	0	0
$901$	−5.88242	−0.195972
$902$	0	0
$903$	42.9740	1.43008
$904$	0	0
$905$	8.95847	0.297790
$906$	0	0
$907$	−46.4475	−1.54226	−0.771132	−	0.636675i	$-0.780309\pi$
$907$	−46.4475	−1.54226	−0.771132	+	0.636675i	$0.780309\pi$
$908$	0	0
$909$	−66.1649	−2.19455
$910$	0	0
$911$	−9.67776	−0.320638	−0.160319	−	0.987065i	$-0.551252\pi$
$911$	−9.67776	−0.320638	−0.160319	+	0.987065i	$0.551252\pi$
$912$	0	0
$913$	4.67515	0.154725
$914$	0	0
$915$	−87.4028	−2.88945
$916$	0	0
$917$	−21.5218	−0.710713
$918$	0	0
$919$	1.82359	0.0601546	0.0300773	−	0.999548i	$-0.490425\pi$
$919$	1.82359	0.0601546	0.0300773	+	0.999548i	$0.490425\pi$
$920$	0	0
$921$	100.543	3.31300
$922$	0	0
$923$	20.5473	0.676323
$924$	0	0
$925$	−34.6041	−1.13777
$926$	0	0
$927$	−149.330	−4.90465
$928$	0	0
$929$	−23.8989	−0.784098	−0.392049	−	0.919944i	$-0.628234\pi$
$929$	−23.8989	−0.784098	−0.392049	+	0.919944i	$0.628234\pi$
$930$	0	0
$931$	5.88776	0.192964
$932$	0	0
$933$	−82.5043	−2.70107
$934$	0	0
$935$	−5.88781	−0.192552
$936$	0	0
$937$	−7.10061	−0.231967	−0.115983	−	0.993251i	$-0.537002\pi$
$937$	−7.10061	−0.231967	−0.115983	+	0.993251i	$0.537002\pi$
$938$	0	0
$939$	53.5976	1.74909
$940$	0	0
$941$	34.6443	1.12937	0.564686	−	0.825306i	$-0.308997\pi$
$941$	34.6443	1.12937	0.564686	+	0.825306i	$0.308997\pi$
$942$	0	0
$943$	24.1095	0.785113
$944$	0	0
$945$	−47.9925	−1.56120
$946$	0	0
$947$	−58.1245	−1.88879	−0.944396	−	0.328810i	$-0.893353\pi$
$947$	−58.1245	−1.88879	−0.944396	+	0.328810i	$0.893353\pi$
$948$	0	0
$949$	−69.8089	−2.26609
$950$	0	0
$951$	73.2585	2.37557
$952$	0	0
$953$	−15.9422	−0.516419	−0.258209	−	0.966089i	$-0.583132\pi$
$953$	−15.9422	−0.516419	−0.258209	+	0.966089i	$0.583132\pi$
$954$	0	0
$955$	−6.85515	−0.221828
$956$	0	0
$957$	0.681535	0.0220309
$958$	0	0
$959$	21.3098	0.688128
$960$	0	0
$961$	−27.9345	−0.901113
$962$	0	0
$963$	135.624	4.37041
$964$	0	0
$965$	25.4468	0.819160
$966$	0	0
$967$	26.4373	0.850167	0.425083	−	0.905154i	$-0.360245\pi$
$967$	26.4373	0.850167	0.425083	+	0.905154i	$0.360245\pi$
$968$	0	0
$969$	−19.1741	−0.615959
$970$	0	0
$971$	−12.5015	−0.401192	−0.200596	−	0.979674i	$-0.564288\pi$
$971$	−12.5015	−0.401192	−0.200596	+	0.979674i	$0.564288\pi$
$972$	0	0
$973$	−16.1684	−0.518337
$974$	0	0
$975$	−79.7730	−2.55478
$976$	0	0
$977$	−43.8965	−1.40437	−0.702187	−	0.711993i	$-0.747793\pi$
$977$	−43.8965	−1.40437	−0.702187	+	0.711993i	$0.747793\pi$
$978$	0	0
$979$	0.251952	0.00805241
$980$	0	0
$981$	−51.2166	−1.63522
$982$	0	0
$983$	61.3680	1.95734	0.978668	−	0.205447i	$-0.0658648\pi$
$983$	61.3680	1.95734	0.978668	+	0.205447i	$0.0658648\pi$
$984$	0	0
$985$	18.1428	0.578079
$986$	0	0
$987$	40.1991	1.27955
$988$	0	0
$989$	87.8276	2.79275
$990$	0	0
$991$	11.7544	0.373392	0.186696	−	0.982418i	$-0.440222\pi$
$991$	11.7544	0.373392	0.186696	+	0.982418i	$0.440222\pi$
$992$	0	0
$993$	85.8848	2.72547
$994$	0	0
$995$	82.5848	2.61812
$996$	0	0
$997$	−28.4239	−0.900193	−0.450097	−	0.892980i	$-0.648610\pi$
$997$	−28.4239	−0.900193	−0.450097	+	0.892980i	$0.648610\pi$
$998$	0	0
$999$	126.818	4.01234

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.1	✓	19	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-3.25955 q^{3} +3.01882 q^{5} +1.05463 q^{7} +7.62467 q^{9} +O(q^{10})\)	\(q-3.25955 q^{3} +3.01882 q^{5} +1.05463 q^{7} +7.62467 q^{9} +0.331559 q^{11} +5.94995 q^{13} -9.83998 q^{15} -5.88242 q^{17} -1.00000 q^{19} -3.43761 q^{21} -7.02557 q^{23} +4.11325 q^{25} -15.0743 q^{27} -0.630623 q^{29} +1.75086 q^{31} -1.08073 q^{33} +3.18372 q^{35} -8.41283 q^{37} -19.3942 q^{39} -3.43168 q^{41} -12.5011 q^{43} +23.0175 q^{45} -11.6939 q^{47} -5.88776 q^{49} +19.1741 q^{51} +1.00000 q^{53} +1.00092 q^{55} +3.25955 q^{57} +5.04111 q^{59} +8.88242 q^{61} +8.04118 q^{63} +17.9618 q^{65} -9.48288 q^{67} +22.9002 q^{69} +3.45336 q^{71} -11.7327 q^{73} -13.4073 q^{75} +0.349671 q^{77} +1.29391 q^{79} +26.2616 q^{81} +14.1005 q^{83} -17.7579 q^{85} +2.05555 q^{87} +0.759900 q^{89} +6.27497 q^{91} -5.70701 q^{93} -3.01882 q^{95} -7.99724 q^{97} +2.52803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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