LMFDB - Embedded newform 4028.2.a.d.1.9

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			1.9
Root			$0.383819$ 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)

$f(q)$	$=$	$q-0.383819 q^{3} -4.10226 q^{5} +0.287206 q^{7} -2.85268 q^{9} +O(q^{10})$	$q-0.383819 q^{3} -4.10226 q^{5} +0.287206 q^{7} -2.85268 q^{9} +0.513939 q^{11} +2.10829 q^{13} +1.57453 q^{15} -1.15435 q^{17} -1.00000 q^{19} -0.110235 q^{21} +8.21568 q^{23} +11.8286 q^{25} +2.24637 q^{27} -0.842400 q^{29} +5.84212 q^{31} -0.197260 q^{33} -1.17820 q^{35} -4.24583 q^{37} -0.809203 q^{39} +3.80488 q^{41} -1.27243 q^{43} +11.7025 q^{45} +5.24622 q^{47} -6.91751 q^{49} +0.443060 q^{51} +1.00000 q^{53} -2.10831 q^{55} +0.383819 q^{57} +3.73857 q^{59} -13.8747 q^{61} -0.819309 q^{63} -8.64876 q^{65} -0.286137 q^{67} -3.15333 q^{69} +3.50584 q^{71} -7.34802 q^{73} -4.54003 q^{75} +0.147607 q^{77} -12.5141 q^{79} +7.69585 q^{81} -9.32244 q^{83} +4.73543 q^{85} +0.323329 q^{87} +2.32232 q^{89} +0.605515 q^{91} -2.24232 q^{93} +4.10226 q^{95} -17.8148 q^{97} -1.46611 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.383819	−0.221598	−0.110799	−	0.993843i	$-0.535341\pi$
$3$	−0.383819	−0.221598	−0.110799	+	0.993843i	$0.535341\pi$
$4$	0	0
$5$	−4.10226	−1.83459	−0.917294	−	0.398211i	$-0.869631\pi$
$5$	−4.10226	−1.83459	−0.917294	+	0.398211i	$0.869631\pi$
$6$	0	0
$7$	0.287206	0.108554	0.0542769	−	0.998526i	$-0.482715\pi$
$7$	0.287206	0.108554	0.0542769	+	0.998526i	$0.482715\pi$
$8$	0	0
$9$	−2.85268	−0.950894
$10$	0	0
$11$	0.513939	0.154959	0.0774793	−	0.996994i	$-0.475313\pi$
$11$	0.513939	0.154959	0.0774793	+	0.996994i	$0.475313\pi$
$12$	0	0
$13$	2.10829	0.584735	0.292367	−	0.956306i	$-0.405557\pi$
$13$	2.10829	0.584735	0.292367	+	0.956306i	$0.405557\pi$
$14$	0	0
$15$	1.57453	0.406541
$16$	0	0
$17$	−1.15435	−0.279970	−0.139985	−	0.990154i	$-0.544705\pi$
$17$	−1.15435	−0.279970	−0.139985	+	0.990154i	$0.544705\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.110235	−0.0240553
$22$	0	0
$23$	8.21568	1.71309	0.856544	−	0.516075i	$-0.172607\pi$
$23$	8.21568	1.71309	0.856544	+	0.516075i	$0.172607\pi$
$24$	0	0
$25$	11.8286	2.36571
$26$	0	0
$27$	2.24637	0.432315
$28$	0	0
$29$	−0.842400	−0.156430	−0.0782149	−	0.996937i	$-0.524922\pi$
$29$	−0.842400	−0.156430	−0.0782149	+	0.996937i	$0.524922\pi$
$30$	0	0
$31$	5.84212	1.04928	0.524638	−	0.851326i	$-0.324201\pi$
$31$	5.84212	1.04928	0.524638	+	0.851326i	$0.324201\pi$
$32$	0	0
$33$	−0.197260	−0.0343385
$34$	0	0
$35$	−1.17820	−0.199151
$36$	0	0
$37$	−4.24583	−0.698011	−0.349005	−	0.937121i	$-0.613481\pi$
$37$	−4.24583	−0.698011	−0.349005	+	0.937121i	$0.613481\pi$
$38$	0	0
$39$	−0.809203	−0.129576
$40$	0	0
$41$	3.80488	0.594222	0.297111	−	0.954843i	$-0.403977\pi$
$41$	3.80488	0.594222	0.297111	+	0.954843i	$0.403977\pi$
$42$	0	0
$43$	−1.27243	−0.194043	−0.0970215	−	0.995282i	$-0.530932\pi$
$43$	−1.27243	−0.194043	−0.0970215	+	0.995282i	$0.530932\pi$
$44$	0	0
$45$	11.7025	1.74450
$46$	0	0
$47$	5.24622	0.765240	0.382620	−	0.923906i	$-0.375022\pi$
$47$	5.24622	0.765240	0.382620	+	0.923906i	$0.375022\pi$
$48$	0	0
$49$	−6.91751	−0.988216
$50$	0	0
$51$	0.443060	0.0620408
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−2.10831	−0.284285
$56$	0	0
$57$	0.383819	0.0508381
$58$	0	0
$59$	3.73857	0.486721	0.243360	−	0.969936i	$-0.421750\pi$
$59$	3.73857	0.486721	0.243360	+	0.969936i	$0.421750\pi$
$60$	0	0
$61$	−13.8747	−1.77647	−0.888236	−	0.459388i	$-0.848069\pi$
$61$	−13.8747	−1.77647	−0.888236	+	0.459388i	$0.848069\pi$
$62$	0	0
$63$	−0.819309	−0.103223
$64$	0	0
$65$	−8.64876	−1.07275
$66$	0	0
$67$	−0.286137	−0.0349572	−0.0174786	−	0.999847i	$-0.505564\pi$
$67$	−0.286137	−0.0349572	−0.0174786	+	0.999847i	$0.505564\pi$
$68$	0	0
$69$	−3.15333	−0.379617
$70$	0	0
$71$	3.50584	0.416066	0.208033	−	0.978122i	$-0.433294\pi$
$71$	3.50584	0.416066	0.208033	+	0.978122i	$0.433294\pi$
$72$	0	0
$73$	−7.34802	−0.860021	−0.430010	−	0.902824i	$-0.641490\pi$
$73$	−7.34802	−0.860021	−0.430010	+	0.902824i	$0.641490\pi$
$74$	0	0
$75$	−4.54003	−0.524237
$76$	0	0
$77$	0.147607	0.0168213
$78$	0	0
$79$	−12.5141	−1.40794	−0.703970	−	0.710229i	$-0.748591\pi$
$79$	−12.5141	−1.40794	−0.703970	+	0.710229i	$0.748591\pi$
$80$	0	0
$81$	7.69585	0.855094
$82$	0	0
$83$	−9.32244	−1.02327	−0.511635	−	0.859203i	$-0.670960\pi$
$83$	−9.32244	−1.02327	−0.511635	+	0.859203i	$0.670960\pi$
$84$	0	0
$85$	4.73543	0.513629
$86$	0	0
$87$	0.323329	0.0346645
$88$	0	0
$89$	2.32232	0.246165	0.123082	−	0.992396i	$-0.460722\pi$
$89$	2.32232	0.246165	0.123082	+	0.992396i	$0.460722\pi$
$90$	0	0
$91$	0.605515	0.0634752
$92$	0	0
$93$	−2.24232	−0.232517
$94$	0	0
$95$	4.10226	0.420883
$96$	0	0
$97$	−17.8148	−1.80882	−0.904409	−	0.426667i	$-0.859688\pi$
$97$	−17.8148	−1.80882	−0.904409	+	0.426667i	$0.859688\pi$
$98$	0	0
$99$	−1.46611	−0.147349
$100$	0	0
$101$	−4.30447	−0.428311	−0.214156	−	0.976800i	$-0.568700\pi$
$101$	−4.30447	−0.428311	−0.214156	+	0.976800i	$0.568700\pi$
$102$	0	0
$103$	12.3918	1.22100	0.610502	−	0.792015i	$-0.290968\pi$
$103$	12.3918	1.22100	0.610502	+	0.792015i	$0.290968\pi$
$104$	0	0
$105$	0.452214	0.0441316
$106$	0	0
$107$	12.9490	1.25183	0.625915	−	0.779891i	$-0.284726\pi$
$107$	12.9490	1.25183	0.625915	+	0.779891i	$0.284726\pi$
$108$	0	0
$109$	−2.92088	−0.279770	−0.139885	−	0.990168i	$-0.544673\pi$
$109$	−2.92088	−0.279770	−0.139885	+	0.990168i	$0.544673\pi$
$110$	0	0
$111$	1.62963	0.154678
$112$	0	0
$113$	−1.71656	−0.161480	−0.0807400	−	0.996735i	$-0.525728\pi$
$113$	−1.71656	−0.161480	−0.0807400	+	0.996735i	$0.525728\pi$
$114$	0	0
$115$	−33.7029	−3.14281
$116$	0	0
$117$	−6.01429	−0.556021
$118$	0	0
$119$	−0.331535	−0.0303918
$120$	0	0
$121$	−10.7359	−0.975988
$122$	0	0
$123$	−1.46039	−0.131679
$124$	0	0
$125$	−28.0125	−2.50552
$126$	0	0
$127$	−16.9095	−1.50047	−0.750237	−	0.661169i	$-0.770061\pi$
$127$	−16.9095	−1.50047	−0.750237	+	0.661169i	$0.770061\pi$
$128$	0	0
$129$	0.488381	0.0429996
$130$	0	0
$131$	22.7062	1.98385	0.991923	−	0.126840i	$-0.0404833\pi$
$131$	22.7062	1.98385	0.991923	+	0.126840i	$0.0404833\pi$
$132$	0	0
$133$	−0.287206	−0.0249040
$134$	0	0
$135$	−9.21521	−0.793119
$136$	0	0
$137$	13.8985	1.18743	0.593715	−	0.804676i	$-0.297661\pi$
$137$	13.8985	1.18743	0.593715	+	0.804676i	$0.297661\pi$
$138$	0	0
$139$	1.30301	0.110520	0.0552600	−	0.998472i	$-0.482401\pi$
$139$	1.30301	0.110520	0.0552600	+	0.998472i	$0.482401\pi$
$140$	0	0
$141$	−2.01360	−0.169576
$142$	0	0
$143$	1.08353	0.0906097
$144$	0	0
$145$	3.45575	0.286984
$146$	0	0
$147$	2.65507	0.218987
$148$	0	0
$149$	−5.57766	−0.456939	−0.228470	−	0.973551i	$-0.573372\pi$
$149$	−5.57766	−0.456939	−0.228470	+	0.973551i	$0.573372\pi$
$150$	0	0
$151$	−9.20322	−0.748948	−0.374474	−	0.927237i	$-0.622177\pi$
$151$	−9.20322	−0.748948	−0.374474	+	0.927237i	$0.622177\pi$
$152$	0	0
$153$	3.29298	0.266222
$154$	0	0
$155$	−23.9659	−1.92499
$156$	0	0
$157$	13.4840	1.07614	0.538070	−	0.842900i	$-0.319154\pi$
$157$	13.4840	1.07614	0.538070	+	0.842900i	$0.319154\pi$
$158$	0	0
$159$	−0.383819	−0.0304388
$160$	0	0
$161$	2.35960	0.185962
$162$	0	0
$163$	1.16173	0.0909938	0.0454969	−	0.998964i	$-0.485513\pi$
$163$	1.16173	0.0909938	0.0454969	+	0.998964i	$0.485513\pi$
$164$	0	0
$165$	0.809212	0.0629970
$166$	0	0
$167$	21.4507	1.65990	0.829952	−	0.557835i	$-0.188368\pi$
$167$	21.4507	1.65990	0.829952	+	0.557835i	$0.188368\pi$
$168$	0	0
$169$	−8.55511	−0.658085
$170$	0	0
$171$	2.85268	0.218150
$172$	0	0
$173$	−7.76575	−0.590419	−0.295209	−	0.955433i	$-0.595389\pi$
$173$	−7.76575	−0.590419	−0.295209	+	0.955433i	$0.595389\pi$
$174$	0	0
$175$	3.39724	0.256807
$176$	0	0
$177$	−1.43494	−0.107856
$178$	0	0
$179$	−24.8470	−1.85715	−0.928575	−	0.371144i	$-0.878966\pi$
$179$	−24.8470	−1.85715	−0.928575	+	0.371144i	$0.878966\pi$
$180$	0	0
$181$	−6.81457	−0.506523	−0.253262	−	0.967398i	$-0.581503\pi$
$181$	−6.81457	−0.506523	−0.253262	+	0.967398i	$0.581503\pi$
$182$	0	0
$183$	5.32537	0.393663
$184$	0	0
$185$	17.4175	1.28056
$186$	0	0
$187$	−0.593264	−0.0433837
$188$	0	0
$189$	0.645172	0.0469294
$190$	0	0
$191$	11.0292	0.798045	0.399022	−	0.916941i	$-0.369350\pi$
$191$	11.0292	0.798045	0.399022	+	0.916941i	$0.369350\pi$
$192$	0	0
$193$	−5.06967	−0.364923	−0.182461	−	0.983213i	$-0.558406\pi$
$193$	−5.06967	−0.364923	−0.182461	+	0.983213i	$0.558406\pi$
$194$	0	0
$195$	3.31956	0.237719
$196$	0	0
$197$	−16.9139	−1.20507	−0.602534	−	0.798093i	$-0.705842\pi$
$197$	−16.9139	−1.20507	−0.602534	+	0.798093i	$0.705842\pi$
$198$	0	0
$199$	0.979105	0.0694069	0.0347035	−	0.999398i	$-0.488951\pi$
$199$	0.979105	0.0694069	0.0347035	+	0.999398i	$0.488951\pi$
$200$	0	0
$201$	0.109825	0.00774646
$202$	0	0
$203$	−0.241943	−0.0169810
$204$	0	0
$205$	−15.6086	−1.09015
$206$	0	0
$207$	−23.4367	−1.62896
$208$	0	0
$209$	−0.513939	−0.0355499
$210$	0	0
$211$	−21.3914	−1.47265	−0.736324	−	0.676629i	$-0.763440\pi$
$211$	−21.3914	−1.47265	−0.736324	+	0.676629i	$0.763440\pi$
$212$	0	0
$213$	−1.34561	−0.0921995
$214$	0	0
$215$	5.21982	0.355989
$216$	0	0
$217$	1.67789	0.113903
$218$	0	0
$219$	2.82031	0.190579
$220$	0	0
$221$	−2.43370	−0.163708
$222$	0	0
$223$	2.96662	0.198660	0.0993299	−	0.995055i	$-0.468330\pi$
$223$	2.96662	0.198660	0.0993299	+	0.995055i	$0.468330\pi$
$224$	0	0
$225$	−33.7431	−2.24954
$226$	0	0
$227$	−5.27779	−0.350299	−0.175150	−	0.984542i	$-0.556041\pi$
$227$	−5.27779	−0.350299	−0.175150	+	0.984542i	$0.556041\pi$
$228$	0	0
$229$	−18.6077	−1.22963	−0.614816	−	0.788671i	$-0.710770\pi$
$229$	−18.6077	−1.22963	−0.614816	+	0.788671i	$0.710770\pi$
$230$	0	0
$231$	−0.0566543	−0.00372758
$232$	0	0
$233$	−16.1633	−1.05889	−0.529447	−	0.848343i	$-0.677600\pi$
$233$	−16.1633	−1.05889	−0.529447	+	0.848343i	$0.677600\pi$
$234$	0	0
$235$	−21.5214	−1.40390
$236$	0	0
$237$	4.80313	0.311997
$238$	0	0
$239$	−6.42425	−0.415550	−0.207775	−	0.978177i	$-0.566622\pi$
$239$	−6.42425	−0.415550	−0.207775	+	0.978177i	$0.566622\pi$
$240$	0	0
$241$	24.1005	1.55245	0.776225	−	0.630456i	$-0.217132\pi$
$241$	24.1005	1.55245	0.776225	+	0.630456i	$0.217132\pi$
$242$	0	0
$243$	−9.69293	−0.621802
$244$	0	0
$245$	28.3774	1.81297
$246$	0	0
$247$	−2.10829	−0.134147
$248$	0	0
$249$	3.57813	0.226755
$250$	0	0
$251$	−0.107684	−0.00679694	−0.00339847	−	0.999994i	$-0.501082\pi$
$251$	−0.107684	−0.00679694	−0.00339847	+	0.999994i	$0.501082\pi$
$252$	0	0
$253$	4.22236	0.265458
$254$	0	0
$255$	−1.81755	−0.113819
$256$	0	0
$257$	20.1942	1.25968	0.629839	−	0.776726i	$-0.283121\pi$
$257$	20.1942	1.25968	0.629839	+	0.776726i	$0.283121\pi$
$258$	0	0
$259$	−1.21943	−0.0757717
$260$	0	0
$261$	2.40310	0.148748
$262$	0	0
$263$	−17.0964	−1.05421	−0.527104	−	0.849801i	$-0.676722\pi$
$263$	−17.0964	−1.05421	−0.527104	+	0.849801i	$0.676722\pi$
$264$	0	0
$265$	−4.10226	−0.252000
$266$	0	0
$267$	−0.891349	−0.0545497
$268$	0	0
$269$	−17.2783	−1.05348	−0.526739	−	0.850027i	$-0.676586\pi$
$269$	−17.2783	−1.05348	−0.526739	+	0.850027i	$0.676586\pi$
$270$	0	0
$271$	−6.40128	−0.388850	−0.194425	−	0.980917i	$-0.562284\pi$
$271$	−6.40128	−0.388850	−0.194425	+	0.980917i	$0.562284\pi$
$272$	0	0
$273$	−0.232408	−0.0140660
$274$	0	0
$275$	6.07916	0.366587
$276$	0	0
$277$	−4.35360	−0.261582	−0.130791	−	0.991410i	$-0.541752\pi$
$277$	−4.35360	−0.261582	−0.130791	+	0.991410i	$0.541752\pi$
$278$	0	0
$279$	−16.6657	−0.997750
$280$	0	0
$281$	−10.4250	−0.621902	−0.310951	−	0.950426i	$-0.600648\pi$
$281$	−10.4250	−0.621902	−0.310951	+	0.950426i	$0.600648\pi$
$282$	0	0
$283$	30.4909	1.81249	0.906247	−	0.422748i	$-0.138934\pi$
$283$	30.4909	1.81249	0.906247	+	0.422748i	$0.138934\pi$
$284$	0	0
$285$	−1.57453	−0.0932669
$286$	0	0
$287$	1.09279	0.0645051
$288$	0	0
$289$	−15.6675	−0.921617
$290$	0	0
$291$	6.83766	0.400831
$292$	0	0
$293$	−25.5499	−1.49264	−0.746322	−	0.665586i	$-0.768182\pi$
$293$	−25.5499	−1.49264	−0.746322	+	0.665586i	$0.768182\pi$
$294$	0	0
$295$	−15.3366	−0.892931
$296$	0	0
$297$	1.15450	0.0669908
$298$	0	0
$299$	17.3210	1.00170
$300$	0	0
$301$	−0.365449	−0.0210641
$302$	0	0
$303$	1.65214	0.0949130
$304$	0	0
$305$	56.9176	3.25909
$306$	0	0
$307$	−30.1906	−1.72307	−0.861534	−	0.507699i	$-0.830496\pi$
$307$	−30.1906	−1.72307	−0.861534	+	0.507699i	$0.830496\pi$
$308$	0	0
$309$	−4.75623	−0.270572
$310$	0	0
$311$	−1.45973	−0.0827734	−0.0413867	−	0.999143i	$-0.513178\pi$
$311$	−1.45973	−0.0827734	−0.0413867	+	0.999143i	$0.513178\pi$
$312$	0	0
$313$	25.4003	1.43571	0.717855	−	0.696192i	$-0.245124\pi$
$313$	25.4003	1.43571	0.717855	+	0.696192i	$0.245124\pi$
$314$	0	0
$315$	3.36102	0.189372
$316$	0	0
$317$	11.5159	0.646797	0.323399	−	0.946263i	$-0.395175\pi$
$317$	11.5159	0.646797	0.323399	+	0.946263i	$0.395175\pi$
$318$	0	0
$319$	−0.432943	−0.0242401
$320$	0	0
$321$	−4.97009	−0.277403
$322$	0	0
$323$	1.15435	0.0642295
$324$	0	0
$325$	24.9380	1.38331
$326$	0	0
$327$	1.12109	0.0619965
$328$	0	0
$329$	1.50675	0.0830697
$330$	0	0
$331$	10.7679	0.591858	0.295929	−	0.955210i	$-0.404371\pi$
$331$	10.7679	0.591858	0.295929	+	0.955210i	$0.404371\pi$
$332$	0	0
$333$	12.1120	0.663735
$334$	0	0
$335$	1.17381	0.0641321
$336$	0	0
$337$	28.1022	1.53082	0.765412	−	0.643541i	$-0.222535\pi$
$337$	28.1022	1.53082	0.765412	+	0.643541i	$0.222535\pi$
$338$	0	0
$339$	0.658847	0.0357837
$340$	0	0
$341$	3.00249	0.162594
$342$	0	0
$343$	−3.99720	−0.215828
$344$	0	0
$345$	12.9358	0.696440
$346$	0	0
$347$	1.65294	0.0887347	0.0443673	−	0.999015i	$-0.485873\pi$
$347$	1.65294	0.0887347	0.0443673	+	0.999015i	$0.485873\pi$
$348$	0	0
$349$	26.0054	1.39204	0.696018	−	0.718024i	$-0.254953\pi$
$349$	26.0054	1.39204	0.696018	+	0.718024i	$0.254953\pi$
$350$	0	0
$351$	4.73601	0.252789
$352$	0	0
$353$	1.62045	0.0862479	0.0431239	−	0.999070i	$-0.486269\pi$
$353$	1.62045	0.0862479	0.0431239	+	0.999070i	$0.486269\pi$
$354$	0	0
$355$	−14.3819	−0.763310
$356$	0	0
$357$	0.127250	0.00673477
$358$	0	0
$359$	20.6462	1.08966	0.544832	−	0.838545i	$-0.316593\pi$
$359$	20.6462	1.08966	0.544832	+	0.838545i	$0.316593\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	4.12063	0.216277
$364$	0	0
$365$	30.1435	1.57778
$366$	0	0
$367$	−12.5293	−0.654023	−0.327012	−	0.945020i	$-0.606042\pi$
$367$	−12.5293	−0.654023	−0.327012	+	0.945020i	$0.606042\pi$
$368$	0	0
$369$	−10.8541	−0.565043
$370$	0	0
$371$	0.287206	0.0149110
$372$	0	0
$373$	−11.0739	−0.573383	−0.286692	−	0.958023i	$-0.592556\pi$
$373$	−11.0739	−0.573383	−0.286692	+	0.958023i	$0.592556\pi$
$374$	0	0
$375$	10.7517	0.555218
$376$	0	0
$377$	−1.77602	−0.0914699
$378$	0	0
$379$	−19.8531	−1.01979	−0.509893	−	0.860238i	$-0.670315\pi$
$379$	−19.8531	−1.01979	−0.509893	+	0.860238i	$0.670315\pi$
$380$	0	0
$381$	6.49019	0.332502
$382$	0	0
$383$	1.74220	0.0890222	0.0445111	−	0.999009i	$-0.485827\pi$
$383$	1.74220	0.0890222	0.0445111	+	0.999009i	$0.485827\pi$
$384$	0	0
$385$	−0.605521	−0.0308602
$386$	0	0
$387$	3.62983	0.184514
$388$	0	0
$389$	−19.4618	−0.986752	−0.493376	−	0.869816i	$-0.664237\pi$
$389$	−19.4618	−0.986752	−0.493376	+	0.869816i	$0.664237\pi$
$390$	0	0
$391$	−9.48373	−0.479613
$392$	0	0
$393$	−8.71506	−0.439617
$394$	0	0
$395$	51.3359	2.58299
$396$	0	0
$397$	0.208797	0.0104792	0.00523960	−	0.999986i	$-0.498332\pi$
$397$	0.208797	0.0104792	0.00523960	+	0.999986i	$0.498332\pi$
$398$	0	0
$399$	0.110235	0.00551867
$400$	0	0
$401$	−30.0175	−1.49900	−0.749501	−	0.662003i	$-0.769707\pi$
$401$	−30.0175	−1.49900	−0.749501	+	0.662003i	$0.769707\pi$
$402$	0	0
$403$	12.3169	0.613548
$404$	0	0
$405$	−31.5704	−1.56875
$406$	0	0
$407$	−2.18210	−0.108163
$408$	0	0
$409$	−3.07405	−0.152002	−0.0760009	−	0.997108i	$-0.524215\pi$
$409$	−3.07405	−0.152002	−0.0760009	+	0.997108i	$0.524215\pi$
$410$	0	0
$411$	−5.33451	−0.263132
$412$	0	0
$413$	1.07374	0.0528354
$414$	0	0
$415$	38.2431	1.87728
$416$	0	0
$417$	−0.500121	−0.0244910
$418$	0	0
$419$	−30.6784	−1.49874	−0.749368	−	0.662153i	$-0.769643\pi$
$419$	−30.6784	−1.49874	−0.749368	+	0.662153i	$0.769643\pi$
$420$	0	0
$421$	19.3709	0.944080	0.472040	−	0.881577i	$-0.343518\pi$
$421$	19.3709	0.944080	0.472040	+	0.881577i	$0.343518\pi$
$422$	0	0
$423$	−14.9658	−0.727663
$424$	0	0
$425$	−13.6542	−0.662328
$426$	0	0
$427$	−3.98490	−0.192843
$428$	0	0
$429$	−0.415881	−0.0200789
$430$	0	0
$431$	−11.6691	−0.562080	−0.281040	−	0.959696i	$-0.590679\pi$
$431$	−11.6691	−0.562080	−0.281040	+	0.959696i	$0.590679\pi$
$432$	0	0
$433$	−12.9461	−0.622152	−0.311076	−	0.950385i	$-0.600689\pi$
$433$	−12.9461	−0.622152	−0.311076	+	0.950385i	$0.600689\pi$
$434$	0	0
$435$	−1.32638	−0.0635951
$436$	0	0
$437$	−8.21568	−0.393009
$438$	0	0
$439$	−11.2382	−0.536370	−0.268185	−	0.963367i	$-0.586424\pi$
$439$	−11.2382	−0.536370	−0.268185	+	0.963367i	$0.586424\pi$
$440$	0	0
$441$	19.7335	0.939689
$442$	0	0
$443$	−5.76155	−0.273740	−0.136870	−	0.990589i	$-0.543704\pi$
$443$	−5.76155	−0.273740	−0.136870	+	0.990589i	$0.543704\pi$
$444$	0	0
$445$	−9.52675	−0.451611
$446$	0	0
$447$	2.14081	0.101257
$448$	0	0
$449$	−7.32440	−0.345660	−0.172830	−	0.984952i	$-0.555291\pi$
$449$	−7.32440	−0.345660	−0.172830	+	0.984952i	$0.555291\pi$
$450$	0	0
$451$	1.95548	0.0920798
$452$	0	0
$453$	3.53237	0.165965
$454$	0	0
$455$	−2.48398	−0.116451
$456$	0	0
$457$	17.6055	0.823548	0.411774	−	0.911286i	$-0.364909\pi$
$457$	17.6055	0.823548	0.411774	+	0.911286i	$0.364909\pi$
$458$	0	0
$459$	−2.59309	−0.121035
$460$	0	0
$461$	17.5982	0.819631	0.409815	−	0.912168i	$-0.365593\pi$
$461$	17.5982	0.819631	0.409815	+	0.912168i	$0.365593\pi$
$462$	0	0
$463$	−20.5584	−0.955429	−0.477715	−	0.878515i	$-0.658535\pi$
$463$	−20.5584	−0.955429	−0.477715	+	0.878515i	$0.658535\pi$
$464$	0	0
$465$	9.19857	0.426573
$466$	0	0
$467$	15.7545	0.729032	0.364516	−	0.931197i	$-0.381234\pi$
$467$	15.7545	0.729032	0.364516	+	0.931197i	$0.381234\pi$
$468$	0	0
$469$	−0.0821805	−0.00379474
$470$	0	0
$471$	−5.17541	−0.238471
$472$	0	0
$473$	−0.653949	−0.0300686
$474$	0	0
$475$	−11.8286	−0.542731
$476$	0	0
$477$	−2.85268	−0.130615
$478$	0	0
$479$	11.9229	0.544771	0.272385	−	0.962188i	$-0.412187\pi$
$479$	11.9229	0.544771	0.272385	+	0.962188i	$0.412187\pi$
$480$	0	0
$481$	−8.95146	−0.408151
$482$	0	0
$483$	−0.905658	−0.0412089
$484$	0	0
$485$	73.0809	3.31843
$486$	0	0
$487$	−33.1036	−1.50007	−0.750033	−	0.661401i	$-0.769962\pi$
$487$	−33.1036	−1.50007	−0.750033	+	0.661401i	$0.769962\pi$
$488$	0	0
$489$	−0.445895	−0.0201641
$490$	0	0
$491$	0.777119	0.0350709	0.0175355	−	0.999846i	$-0.494418\pi$
$491$	0.777119	0.0350709	0.0175355	+	0.999846i	$0.494418\pi$
$492$	0	0
$493$	0.972421	0.0437956
$494$	0	0
$495$	6.01435	0.270325
$496$	0	0
$497$	1.00690	0.0451656
$498$	0	0
$499$	−9.81805	−0.439516	−0.219758	−	0.975554i	$-0.570527\pi$
$499$	−9.81805	−0.439516	−0.219758	+	0.975554i	$0.570527\pi$
$500$	0	0
$501$	−8.23318	−0.367831
$502$	0	0
$503$	−3.49990	−0.156053	−0.0780263	−	0.996951i	$-0.524862\pi$
$503$	−3.49990	−0.156053	−0.0780263	+	0.996951i	$0.524862\pi$
$504$	0	0
$505$	17.6581	0.785774
$506$	0	0
$507$	3.28361	0.145830
$508$	0	0
$509$	−29.5525	−1.30989	−0.654946	−	0.755676i	$-0.727309\pi$
$509$	−29.5525	−1.30989	−0.654946	+	0.755676i	$0.727309\pi$
$510$	0	0
$511$	−2.11040	−0.0933585
$512$	0	0
$513$	−2.24637	−0.0991798
$514$	0	0
$515$	−50.8346	−2.24004
$516$	0	0
$517$	2.69624	0.118581
$518$	0	0
$519$	2.98064	0.130836
$520$	0	0
$521$	2.32943	0.102054	0.0510270	−	0.998697i	$-0.483751\pi$
$521$	2.32943	0.102054	0.0510270	+	0.998697i	$0.483751\pi$
$522$	0	0
$523$	−11.2617	−0.492440	−0.246220	−	0.969214i	$-0.579189\pi$
$523$	−11.2617	−0.492440	−0.246220	+	0.969214i	$0.579189\pi$
$524$	0	0
$525$	−1.30392	−0.0569079
$526$	0	0
$527$	−6.74382	−0.293765
$528$	0	0
$529$	44.4974	1.93467
$530$	0	0
$531$	−10.6650	−0.462820
$532$	0	0
$533$	8.02179	0.347462
$534$	0	0
$535$	−53.1203	−2.29659
$536$	0	0
$537$	9.53675	0.411541
$538$	0	0
$539$	−3.55518	−0.153133
$540$	0	0
$541$	30.8328	1.32561	0.662803	−	0.748794i	$-0.269367\pi$
$541$	30.8328	1.32561	0.662803	+	0.748794i	$0.269367\pi$
$542$	0	0
$543$	2.61556	0.112245
$544$	0	0
$545$	11.9822	0.513263
$546$	0	0
$547$	10.2865	0.439818	0.219909	−	0.975520i	$-0.429424\pi$
$547$	10.2865	0.439818	0.219909	+	0.975520i	$0.429424\pi$
$548$	0	0
$549$	39.5801	1.68924
$550$	0	0
$551$	0.842400	0.0358874
$552$	0	0
$553$	−3.59412	−0.152837
$554$	0	0
$555$	−6.68518	−0.283770
$556$	0	0
$557$	−29.3582	−1.24395	−0.621974	−	0.783038i	$-0.713669\pi$
$557$	−29.3582	−1.24395	−0.621974	+	0.783038i	$0.713669\pi$
$558$	0	0
$559$	−2.68264	−0.113464
$560$	0	0
$561$	0.227706	0.00961376
$562$	0	0
$563$	−12.1101	−0.510380	−0.255190	−	0.966891i	$-0.582138\pi$
$563$	−12.1101	−0.510380	−0.255190	+	0.966891i	$0.582138\pi$
$564$	0	0
$565$	7.04176	0.296249
$566$	0	0
$567$	2.21030	0.0928237
$568$	0	0
$569$	23.2912	0.976417	0.488208	−	0.872727i	$-0.337651\pi$
$569$	23.2912	0.976417	0.488208	+	0.872727i	$0.337651\pi$
$570$	0	0
$571$	5.01580	0.209905	0.104952	−	0.994477i	$-0.466531\pi$
$571$	5.01580	0.209905	0.104952	+	0.994477i	$0.466531\pi$
$572$	0	0
$573$	−4.23322	−0.176845
$574$	0	0
$575$	97.1796	4.05267
$576$	0	0
$577$	13.2297	0.550760	0.275380	−	0.961335i	$-0.411196\pi$
$577$	13.2297	0.550760	0.275380	+	0.961335i	$0.411196\pi$
$578$	0	0
$579$	1.94584	0.0808662
$580$	0	0
$581$	−2.67746	−0.111080
$582$	0	0
$583$	0.513939	0.0212852
$584$	0	0
$585$	24.6722	1.02007
$586$	0	0
$587$	16.5555	0.683319	0.341659	−	0.939824i	$-0.389011\pi$
$587$	16.5555	0.683319	0.341659	+	0.939824i	$0.389011\pi$
$588$	0	0
$589$	−5.84212	−0.240720
$590$	0	0
$591$	6.49189	0.267041
$592$	0	0
$593$	13.4410	0.551957	0.275979	−	0.961164i	$-0.410998\pi$
$593$	13.4410	0.551957	0.275979	+	0.961164i	$0.410998\pi$
$594$	0	0
$595$	1.36005	0.0557564
$596$	0	0
$597$	−0.375799	−0.0153804
$598$	0	0
$599$	−3.22531	−0.131783	−0.0658913	−	0.997827i	$-0.520989\pi$
$599$	−3.22531	−0.131783	−0.0658913	+	0.997827i	$0.520989\pi$
$600$	0	0
$601$	−4.21383	−0.171886	−0.0859429	−	0.996300i	$-0.527390\pi$
$601$	−4.21383	−0.171886	−0.0859429	+	0.996300i	$0.527390\pi$
$602$	0	0
$603$	0.816259	0.0332406
$604$	0	0
$605$	44.0413	1.79053
$606$	0	0
$607$	−17.4099	−0.706646	−0.353323	−	0.935501i	$-0.614948\pi$
$607$	−17.4099	−0.706646	−0.353323	+	0.935501i	$0.614948\pi$
$608$	0	0
$609$	0.0928622	0.00376297
$610$	0	0
$611$	11.0606	0.447463
$612$	0	0
$613$	2.92550	0.118160	0.0590800	−	0.998253i	$-0.481183\pi$
$613$	2.92550	0.118160	0.0590800	+	0.998253i	$0.481183\pi$
$614$	0	0
$615$	5.99089	0.241576
$616$	0	0
$617$	−13.3011	−0.535484	−0.267742	−	0.963491i	$-0.586277\pi$
$617$	−13.3011	−0.535484	−0.267742	+	0.963491i	$0.586277\pi$
$618$	0	0
$619$	−28.7135	−1.15409	−0.577047	−	0.816711i	$-0.695795\pi$
$619$	−28.7135	−1.15409	−0.577047	+	0.816711i	$0.695795\pi$
$620$	0	0
$621$	18.4555	0.740592
$622$	0	0
$623$	0.666984	0.0267221
$624$	0	0
$625$	55.7719	2.23088
$626$	0	0
$627$	0.197260	0.00787780
$628$	0	0
$629$	4.90116	0.195422
$630$	0	0
$631$	−25.0365	−0.996686	−0.498343	−	0.866980i	$-0.666058\pi$
$631$	−25.0365	−0.996686	−0.498343	+	0.866980i	$0.666058\pi$
$632$	0	0
$633$	8.21045	0.326336
$634$	0	0
$635$	69.3671	2.75275
$636$	0	0
$637$	−14.5841	−0.577844
$638$	0	0
$639$	−10.0010	−0.395635
$640$	0	0
$641$	−14.9778	−0.591589	−0.295795	−	0.955252i	$-0.595584\pi$
$641$	−14.9778	−0.591589	−0.295795	+	0.955252i	$0.595584\pi$
$642$	0	0
$643$	−20.9750	−0.827172	−0.413586	−	0.910465i	$-0.635724\pi$
$643$	−20.9750	−0.827172	−0.413586	+	0.910465i	$0.635724\pi$
$644$	0	0
$645$	−2.00347	−0.0788865
$646$	0	0
$647$	−34.5495	−1.35828	−0.679140	−	0.734009i	$-0.737647\pi$
$647$	−34.5495	−1.35828	−0.679140	+	0.734009i	$0.737647\pi$
$648$	0	0
$649$	1.92140	0.0754215
$650$	0	0
$651$	−0.644008	−0.0252406
$652$	0	0
$653$	27.9690	1.09451	0.547256	−	0.836965i	$-0.315673\pi$
$653$	27.9690	1.09451	0.547256	+	0.836965i	$0.315673\pi$
$654$	0	0
$655$	−93.1466	−3.63954
$656$	0	0
$657$	20.9616	0.817789
$658$	0	0
$659$	20.7208	0.807169	0.403584	−	0.914942i	$-0.367764\pi$
$659$	20.7208	0.807169	0.403584	+	0.914942i	$0.367764\pi$
$660$	0	0
$661$	29.5669	1.15002	0.575010	−	0.818146i	$-0.304998\pi$
$661$	29.5669	1.15002	0.575010	+	0.818146i	$0.304998\pi$
$662$	0	0
$663$	0.934100	0.0362774
$664$	0	0
$665$	1.17820	0.0456885
$666$	0	0
$667$	−6.92089	−0.267978
$668$	0	0
$669$	−1.13865	−0.0440226
$670$	0	0
$671$	−7.13075	−0.275279
$672$	0	0
$673$	31.3600	1.20884	0.604420	−	0.796666i	$-0.293405\pi$
$673$	31.3600	1.20884	0.604420	+	0.796666i	$0.293405\pi$
$674$	0	0
$675$	26.5713	1.02273
$676$	0	0
$677$	−43.3889	−1.66757	−0.833785	−	0.552089i	$-0.813831\pi$
$677$	−43.3889	−1.66757	−0.833785	+	0.552089i	$0.813831\pi$
$678$	0	0
$679$	−5.11652	−0.196354
$680$	0	0
$681$	2.02572	0.0776257
$682$	0	0
$683$	17.2824	0.661292	0.330646	−	0.943755i	$-0.392733\pi$
$683$	17.2824	0.661292	0.330646	+	0.943755i	$0.392733\pi$
$684$	0	0
$685$	−57.0153	−2.17844
$686$	0	0
$687$	7.14200	0.272484
$688$	0	0
$689$	2.10829	0.0803195
$690$	0	0
$691$	−15.1753	−0.577296	−0.288648	−	0.957435i	$-0.593206\pi$
$691$	−15.1753	−0.577296	−0.288648	+	0.957435i	$0.593206\pi$
$692$	0	0
$693$	−0.421075	−0.0159953
$694$	0	0
$695$	−5.34530	−0.202759
$696$	0	0
$697$	−4.39215	−0.166364
$698$	0	0
$699$	6.20379	0.234649
$700$	0	0
$701$	38.6222	1.45874	0.729370	−	0.684119i	$-0.239813\pi$
$701$	38.6222	1.45874	0.729370	+	0.684119i	$0.239813\pi$
$702$	0	0
$703$	4.24583	0.160135
$704$	0	0
$705$	8.26032	0.311102
$706$	0	0
$707$	−1.23627	−0.0464948
$708$	0	0
$709$	−52.3288	−1.96525	−0.982625	−	0.185605i	$-0.940576\pi$
$709$	−52.3288	−1.96525	−0.982625	+	0.185605i	$0.940576\pi$
$710$	0	0
$711$	35.6986	1.33880
$712$	0	0
$713$	47.9969	1.79750
$714$	0	0
$715$	−4.44494	−0.166231
$716$	0	0
$717$	2.46575	0.0920851
$718$	0	0
$719$	23.4900	0.876029	0.438014	−	0.898968i	$-0.355682\pi$
$719$	23.4900	0.876029	0.438014	+	0.898968i	$0.355682\pi$
$720$	0	0
$721$	3.55901	0.132545
$722$	0	0
$723$	−9.25024	−0.344020
$724$	0	0
$725$	−9.96437	−0.370068
$726$	0	0
$727$	−23.7890	−0.882286	−0.441143	−	0.897437i	$-0.645427\pi$
$727$	−23.7890	−0.882286	−0.441143	+	0.897437i	$0.645427\pi$
$728$	0	0
$729$	−19.3672	−0.717304
$730$	0	0
$731$	1.46882	0.0543262
$732$	0	0
$733$	4.04253	0.149314	0.0746571	−	0.997209i	$-0.476214\pi$
$733$	4.04253	0.149314	0.0746571	+	0.997209i	$0.476214\pi$
$734$	0	0
$735$	−10.8918	−0.401750
$736$	0	0
$737$	−0.147057	−0.00541692
$738$	0	0
$739$	48.5927	1.78751	0.893757	−	0.448552i	$-0.148060\pi$
$739$	48.5927	1.78751	0.893757	+	0.448552i	$0.148060\pi$
$740$	0	0
$741$	0.809203	0.0297268
$742$	0	0
$743$	−12.7592	−0.468090	−0.234045	−	0.972226i	$-0.575196\pi$
$743$	−12.7592	−0.468090	−0.234045	+	0.972226i	$0.575196\pi$
$744$	0	0
$745$	22.8810	0.838295
$746$	0	0
$747$	26.5940	0.973022
$748$	0	0
$749$	3.71905	0.135891
$750$	0	0
$751$	−27.0166	−0.985851	−0.492926	−	0.870071i	$-0.664073\pi$
$751$	−27.0166	−0.985851	−0.492926	+	0.870071i	$0.664073\pi$
$752$	0	0
$753$	0.0413311	0.00150619
$754$	0	0
$755$	37.7540	1.37401
$756$	0	0
$757$	22.9827	0.835320	0.417660	−	0.908603i	$-0.362850\pi$
$757$	22.9827	0.835320	0.417660	+	0.908603i	$0.362850\pi$
$758$	0	0
$759$	−1.62062	−0.0588249
$760$	0	0
$761$	−4.39314	−0.159251	−0.0796256	−	0.996825i	$-0.525372\pi$
$761$	−4.39314	−0.159251	−0.0796256	+	0.996825i	$0.525372\pi$
$762$	0	0
$763$	−0.838897	−0.0303701
$764$	0	0
$765$	−13.5087	−0.488407
$766$	0	0
$767$	7.88200	0.284602
$768$	0	0
$769$	7.56514	0.272806	0.136403	−	0.990653i	$-0.456446\pi$
$769$	7.56514	0.272806	0.136403	+	0.990653i	$0.456446\pi$
$770$	0	0
$771$	−7.75091	−0.279142
$772$	0	0
$773$	−39.9962	−1.43856	−0.719282	−	0.694718i	$-0.755529\pi$
$773$	−39.9962	−1.43856	−0.719282	+	0.694718i	$0.755529\pi$
$774$	0	0
$775$	69.1038	2.48228
$776$	0	0
$777$	0.468041	0.0167909
$778$	0	0
$779$	−3.80488	−0.136324
$780$	0	0
$781$	1.80179	0.0644730
$782$	0	0
$783$	−1.89234	−0.0676269
$784$	0	0
$785$	−55.3149	−1.97427
$786$	0	0
$787$	19.1406	0.682289	0.341145	−	0.940011i	$-0.389185\pi$
$787$	19.1406	0.682289	0.341145	+	0.940011i	$0.389185\pi$
$788$	0	0
$789$	6.56192	0.233610
$790$	0	0
$791$	−0.493006	−0.0175293
$792$	0	0
$793$	−29.2519	−1.03876
$794$	0	0
$795$	1.57453	0.0558427
$796$	0	0
$797$	−45.9035	−1.62599	−0.812994	−	0.582273i	$-0.802164\pi$
$797$	−45.9035	−1.62599	−0.812994	+	0.582273i	$0.802164\pi$
$798$	0	0
$799$	−6.05595	−0.214244
$800$	0	0
$801$	−6.62483	−0.234077
$802$	0	0
$803$	−3.77644	−0.133268
$804$	0	0
$805$	−9.67968	−0.341164
$806$	0	0
$807$	6.63175	0.233449
$808$	0	0
$809$	−8.78468	−0.308853	−0.154426	−	0.988004i	$-0.549353\pi$
$809$	−8.78468	−0.308853	−0.154426	+	0.988004i	$0.549353\pi$
$810$	0	0
$811$	−23.8294	−0.836763	−0.418382	−	0.908271i	$-0.637403\pi$
$811$	−23.8294	−0.836763	−0.418382	+	0.908271i	$0.637403\pi$
$812$	0	0
$813$	2.45694	0.0861685
$814$	0	0
$815$	−4.76572	−0.166936
$816$	0	0
$817$	1.27243	0.0445165
$818$	0	0
$819$	−1.72734	−0.0603582
$820$	0	0
$821$	12.4667	0.435089	0.217545	−	0.976050i	$-0.430195\pi$
$821$	12.4667	0.435089	0.217545	+	0.976050i	$0.430195\pi$
$822$	0	0
$823$	−18.8253	−0.656207	−0.328104	−	0.944642i	$-0.606410\pi$
$823$	−18.8253	−0.656207	−0.328104	+	0.944642i	$0.606410\pi$
$824$	0	0
$825$	−2.33330	−0.0812350
$826$	0	0
$827$	−20.6971	−0.719709	−0.359854	−	0.933008i	$-0.617174\pi$
$827$	−20.6971	−0.719709	−0.359854	+	0.933008i	$0.617174\pi$
$828$	0	0
$829$	19.3184	0.670955	0.335477	−	0.942048i	$-0.391102\pi$
$829$	19.3184	0.670955	0.335477	+	0.942048i	$0.391102\pi$
$830$	0	0
$831$	1.67100	0.0579662
$832$	0	0
$833$	7.98520	0.276671
$834$	0	0
$835$	−87.9963	−3.04524
$836$	0	0
$837$	13.1236	0.453617
$838$	0	0
$839$	−8.48295	−0.292864	−0.146432	−	0.989221i	$-0.546779\pi$
$839$	−8.48295	−0.292864	−0.146432	+	0.989221i	$0.546779\pi$
$840$	0	0
$841$	−28.2904	−0.975530
$842$	0	0
$843$	4.00131	0.137812
$844$	0	0
$845$	35.0953	1.20731
$846$	0	0
$847$	−3.08341	−0.105947
$848$	0	0
$849$	−11.7030	−0.401645
$850$	0	0
$851$	−34.8824	−1.19575
$852$	0	0
$853$	11.3568	0.388851	0.194425	−	0.980917i	$-0.437716\pi$
$853$	11.3568	0.388851	0.194425	+	0.980917i	$0.437716\pi$
$854$	0	0
$855$	−11.7025	−0.400215
$856$	0	0
$857$	−18.6348	−0.636553	−0.318277	−	0.947998i	$-0.603104\pi$
$857$	−18.6348	−0.636553	−0.318277	+	0.947998i	$0.603104\pi$
$858$	0	0
$859$	−33.3031	−1.13629	−0.568144	−	0.822929i	$-0.692338\pi$
$859$	−33.3031	−1.13629	−0.568144	+	0.822929i	$0.692338\pi$
$860$	0	0
$861$	−0.419432	−0.0142942
$862$	0	0
$863$	−5.03616	−0.171433	−0.0857165	−	0.996320i	$-0.527318\pi$
$863$	−5.03616	−0.171433	−0.0857165	+	0.996320i	$0.527318\pi$
$864$	0	0
$865$	31.8571	1.08317
$866$	0	0
$867$	6.01348	0.204229
$868$	0	0
$869$	−6.43147	−0.218173
$870$	0	0
$871$	−0.603261	−0.0204407
$872$	0	0
$873$	50.8199	1.71999
$874$	0	0
$875$	−8.04537	−0.271983
$876$	0	0
$877$	−56.1751	−1.89690	−0.948450	−	0.316928i	$-0.897349\pi$
$877$	−56.1751	−1.89690	−0.948450	+	0.316928i	$0.897349\pi$
$878$	0	0
$879$	9.80656	0.330767
$880$	0	0
$881$	39.6428	1.33560	0.667801	−	0.744340i	$-0.267236\pi$
$881$	39.6428	1.33560	0.667801	+	0.744340i	$0.267236\pi$
$882$	0	0
$883$	25.5965	0.861390	0.430695	−	0.902497i	$-0.358268\pi$
$883$	25.5965	0.861390	0.430695	+	0.902497i	$0.358268\pi$
$884$	0	0
$885$	5.88648	0.197872
$886$	0	0
$887$	−2.94753	−0.0989685	−0.0494843	−	0.998775i	$-0.515758\pi$
$887$	−2.94753	−0.0989685	−0.0494843	+	0.998775i	$0.515758\pi$
$888$	0	0
$889$	−4.85651	−0.162882
$890$	0	0
$891$	3.95520	0.132504
$892$	0	0
$893$	−5.24622	−0.175558
$894$	0	0
$895$	101.929	3.40711
$896$	0	0
$897$	−6.64815	−0.221975
$898$	0	0
$899$	−4.92140	−0.164138
$900$	0	0
$901$	−1.15435	−0.0384568
$902$	0	0
$903$	0.140266	0.00466777
$904$	0	0
$905$	27.9552	0.929261
$906$	0	0
$907$	41.6606	1.38332	0.691659	−	0.722224i	$-0.256880\pi$
$907$	41.6606	1.38332	0.691659	+	0.722224i	$0.256880\pi$
$908$	0	0
$909$	12.2793	0.407279
$910$	0	0
$911$	−40.7596	−1.35043	−0.675213	−	0.737623i	$-0.735948\pi$
$911$	−40.7596	−1.35043	−0.675213	+	0.737623i	$0.735948\pi$
$912$	0	0
$913$	−4.79117	−0.158565
$914$	0	0
$915$	−21.8461	−0.722209
$916$	0	0
$917$	6.52135	0.215354
$918$	0	0
$919$	32.3314	1.06652	0.533258	−	0.845953i	$-0.320968\pi$
$919$	32.3314	1.06652	0.533258	+	0.845953i	$0.320968\pi$
$920$	0	0
$921$	11.5877	0.381829
$922$	0	0
$923$	7.39133	0.243288
$924$	0	0
$925$	−50.2221	−1.65129
$926$	0	0
$927$	−35.3500	−1.16105
$928$	0	0
$929$	39.0707	1.28187	0.640934	−	0.767596i	$-0.278547\pi$
$929$	39.0707	1.28187	0.640934	+	0.767596i	$0.278547\pi$
$930$	0	0
$931$	6.91751	0.226712
$932$	0	0
$933$	0.560271	0.0183424
$934$	0	0
$935$	2.43372	0.0795913
$936$	0	0
$937$	47.3450	1.54669	0.773347	−	0.633983i	$-0.218581\pi$
$937$	47.3450	1.54669	0.773347	+	0.633983i	$0.218581\pi$
$938$	0	0
$939$	−9.74913	−0.318151
$940$	0	0
$941$	−28.7459	−0.937088	−0.468544	−	0.883440i	$-0.655221\pi$
$941$	−28.7459	−0.937088	−0.468544	+	0.883440i	$0.655221\pi$
$942$	0	0
$943$	31.2597	1.01795
$944$	0	0
$945$	−2.64667	−0.0860961
$946$	0	0
$947$	−31.0986	−1.01057	−0.505284	−	0.862953i	$-0.668612\pi$
$947$	−31.0986	−1.01057	−0.505284	+	0.862953i	$0.668612\pi$
$948$	0	0
$949$	−15.4918	−0.502884
$950$	0	0
$951$	−4.42002	−0.143329
$952$	0	0
$953$	7.70006	0.249429	0.124715	−	0.992193i	$-0.460198\pi$
$953$	7.70006	0.249429	0.124715	+	0.992193i	$0.460198\pi$
$954$	0	0
$955$	−45.2447	−1.46408
$956$	0	0
$957$	0.166172	0.00537157
$958$	0	0
$959$	3.99174	0.128900
$960$	0	0
$961$	3.13032	0.100978
$962$	0	0
$963$	−36.9395	−1.19036
$964$	0	0
$965$	20.7971	0.669482
$966$	0	0
$967$	−54.0601	−1.73845	−0.869227	−	0.494413i	$-0.835383\pi$
$967$	−54.0601	−1.73845	−0.869227	+	0.494413i	$0.835383\pi$
$968$	0	0
$969$	−0.443060	−0.0142331
$970$	0	0
$971$	−15.7172	−0.504388	−0.252194	−	0.967677i	$-0.581152\pi$
$971$	−15.7172	−0.504388	−0.252194	+	0.967677i	$0.581152\pi$
$972$	0	0
$973$	0.374234	0.0119974
$974$	0	0
$975$	−9.57170	−0.306540
$976$	0	0
$977$	−45.6254	−1.45969	−0.729843	−	0.683615i	$-0.760407\pi$
$977$	−45.6254	−1.45969	−0.729843	+	0.683615i	$0.760407\pi$
$978$	0	0
$979$	1.19353	0.0381454
$980$	0	0
$981$	8.33236	0.266032
$982$	0	0
$983$	46.4886	1.48276	0.741379	−	0.671087i	$-0.234172\pi$
$983$	46.4886	1.48276	0.741379	+	0.671087i	$0.234172\pi$
$984$	0	0
$985$	69.3854	2.21080
$986$	0	0
$987$	−0.578319	−0.0184081
$988$	0	0
$989$	−10.4538	−0.332413
$990$	0	0
$991$	−4.45801	−0.141613	−0.0708067	−	0.997490i	$-0.522557\pi$
$991$	−4.45801	−0.141613	−0.0708067	+	0.997490i	$0.522557\pi$
$992$	0	0
$993$	−4.13293	−0.131155
$994$	0	0
$995$	−4.01655	−0.127333
$996$	0	0
$997$	6.55199	0.207503	0.103752	−	0.994603i	$-0.466915\pi$
$997$	6.55199	0.207503	0.103752	+	0.994603i	$0.466915\pi$
$998$	0	0
$999$	−9.53773	−0.301760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.9	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.9	✓	19	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-0.383819 q^{3} -4.10226 q^{5} +0.287206 q^{7} -2.85268 q^{9} +O(q^{10})\)	\(q-0.383819 q^{3} -4.10226 q^{5} +0.287206 q^{7} -2.85268 q^{9} +0.513939 q^{11} +2.10829 q^{13} +1.57453 q^{15} -1.15435 q^{17} -1.00000 q^{19} -0.110235 q^{21} +8.21568 q^{23} +11.8286 q^{25} +2.24637 q^{27} -0.842400 q^{29} +5.84212 q^{31} -0.197260 q^{33} -1.17820 q^{35} -4.24583 q^{37} -0.809203 q^{39} +3.80488 q^{41} -1.27243 q^{43} +11.7025 q^{45} +5.24622 q^{47} -6.91751 q^{49} +0.443060 q^{51} +1.00000 q^{53} -2.10831 q^{55} +0.383819 q^{57} +3.73857 q^{59} -13.8747 q^{61} -0.819309 q^{63} -8.64876 q^{65} -0.286137 q^{67} -3.15333 q^{69} +3.50584 q^{71} -7.34802 q^{73} -4.54003 q^{75} +0.147607 q^{77} -12.5141 q^{79} +7.69585 q^{81} -9.32244 q^{83} +4.73543 q^{85} +0.323329 q^{87} +2.32232 q^{89} +0.605515 q^{91} -2.24232 q^{93} +4.10226 q^{95} -17.8148 q^{97} -1.46611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 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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