LMFDB - Embedded newform 4028.2.a.d.1.19

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			1.19
Root			$-3.17454$ 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)

$f(q)$	$=$	$q+3.17454 q^{3} -4.27638 q^{5} -4.73492 q^{7} +7.07773 q^{9} +O(q^{10})$	$q+3.17454 q^{3} -4.27638 q^{5} -4.73492 q^{7} +7.07773 q^{9} +3.63629 q^{11} +3.74027 q^{13} -13.5756 q^{15} -1.80496 q^{17} -1.00000 q^{19} -15.0312 q^{21} -2.48625 q^{23} +13.2874 q^{25} +12.9449 q^{27} +0.417081 q^{29} -6.42779 q^{31} +11.5436 q^{33} +20.2483 q^{35} -4.97711 q^{37} +11.8737 q^{39} -0.0129771 q^{41} -11.1876 q^{43} -30.2671 q^{45} -6.11779 q^{47} +15.4195 q^{49} -5.72994 q^{51} +1.00000 q^{53} -15.5502 q^{55} -3.17454 q^{57} -12.4760 q^{59} +4.40643 q^{61} -33.5125 q^{63} -15.9948 q^{65} -14.2338 q^{67} -7.89271 q^{69} +4.24339 q^{71} -10.7983 q^{73} +42.1815 q^{75} -17.2176 q^{77} -3.93332 q^{79} +19.8611 q^{81} -11.0579 q^{83} +7.71872 q^{85} +1.32404 q^{87} +3.91713 q^{89} -17.7099 q^{91} -20.4053 q^{93} +4.27638 q^{95} +7.08577 q^{97} +25.7367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.17454	1.83282	0.916412	−	0.400236i	$-0.131072\pi$
$3$	3.17454	1.83282	0.916412	+	0.400236i	$0.131072\pi$
$4$	0	0
$5$	−4.27638	−1.91246	−0.956228	−	0.292623i	$-0.905472\pi$
$5$	−4.27638	−1.91246	−0.956228	+	0.292623i	$0.905472\pi$
$6$	0	0
$7$	−4.73492	−1.78963	−0.894815	−	0.446436i	$-0.852693\pi$
$7$	−4.73492	−1.78963	−0.894815	+	0.446436i	$0.852693\pi$
$8$	0	0
$9$	7.07773	2.35924
$10$	0	0
$11$	3.63629	1.09638	0.548192	−	0.836352i	$-0.315316\pi$
$11$	3.63629	1.09638	0.548192	+	0.836352i	$0.315316\pi$
$12$	0	0
$13$	3.74027	1.03736	0.518682	−	0.854967i	$-0.326423\pi$
$13$	3.74027	1.03736	0.518682	+	0.854967i	$0.326423\pi$
$14$	0	0
$15$	−13.5756	−3.50519
$16$	0	0
$17$	−1.80496	−0.437768	−0.218884	−	0.975751i	$-0.570242\pi$
$17$	−1.80496	−0.437768	−0.218884	+	0.975751i	$0.570242\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−15.0312	−3.28008
$22$	0	0
$23$	−2.48625	−0.518419	−0.259210	−	0.965821i	$-0.583462\pi$
$23$	−2.48625	−0.518419	−0.259210	+	0.965821i	$0.583462\pi$
$24$	0	0
$25$	13.2874	2.65749
$26$	0	0
$27$	12.9449	2.49126
$28$	0	0
$29$	0.417081	0.0774499	0.0387250	−	0.999250i	$-0.487670\pi$
$29$	0.417081	0.0774499	0.0387250	+	0.999250i	$0.487670\pi$
$30$	0	0
$31$	−6.42779	−1.15447	−0.577233	−	0.816580i	$-0.695867\pi$
$31$	−6.42779	−1.15447	−0.577233	+	0.816580i	$0.695867\pi$
$32$	0	0
$33$	11.5436	2.00948
$34$	0	0
$35$	20.2483	3.42259
$36$	0	0
$37$	−4.97711	−0.818231	−0.409116	−	0.912483i	$-0.634163\pi$
$37$	−4.97711	−0.818231	−0.409116	+	0.912483i	$0.634163\pi$
$38$	0	0
$39$	11.8737	1.90131
$40$	0	0
$41$	−0.0129771	−0.00202668	−0.00101334	−	0.999999i	$-0.500323\pi$
$41$	−0.0129771	−0.00202668	−0.00101334	+	0.999999i	$0.500323\pi$
$42$	0	0
$43$	−11.1876	−1.70609	−0.853044	−	0.521839i	$-0.825246\pi$
$43$	−11.1876	−1.70609	−0.853044	+	0.521839i	$0.825246\pi$
$44$	0	0
$45$	−30.2671	−4.51195
$46$	0	0
$47$	−6.11779	−0.892372	−0.446186	−	0.894940i	$-0.647218\pi$
$47$	−6.11779	−0.892372	−0.446186	+	0.894940i	$0.647218\pi$
$48$	0	0
$49$	15.4195	2.20278
$50$	0	0
$51$	−5.72994	−0.802352
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−15.5502	−2.09679
$56$	0	0
$57$	−3.17454	−0.420479
$58$	0	0
$59$	−12.4760	−1.62424	−0.812120	−	0.583491i	$-0.801686\pi$
$59$	−12.4760	−1.62424	−0.812120	+	0.583491i	$0.801686\pi$
$60$	0	0
$61$	4.40643	0.564186	0.282093	−	0.959387i	$-0.408971\pi$
$61$	4.40643	0.564186	0.282093	+	0.959387i	$0.408971\pi$
$62$	0	0
$63$	−33.5125	−4.22218
$64$	0	0
$65$	−15.9948	−1.98391
$66$	0	0
$67$	−14.2338	−1.73894	−0.869470	−	0.493986i	$-0.835539\pi$
$67$	−14.2338	−1.73894	−0.869470	+	0.493986i	$0.835539\pi$
$68$	0	0
$69$	−7.89271	−0.950171
$70$	0	0
$71$	4.24339	0.503597	0.251799	−	0.967780i	$-0.418978\pi$
$71$	4.24339	0.503597	0.251799	+	0.967780i	$0.418978\pi$
$72$	0	0
$73$	−10.7983	−1.26384	−0.631921	−	0.775033i	$-0.717733\pi$
$73$	−10.7983	−1.26384	−0.631921	+	0.775033i	$0.717733\pi$
$74$	0	0
$75$	42.1815	4.87071
$76$	0	0
$77$	−17.2176	−1.96212
$78$	0	0
$79$	−3.93332	−0.442533	−0.221267	−	0.975213i	$-0.571019\pi$
$79$	−3.93332	−0.442533	−0.221267	+	0.975213i	$0.571019\pi$
$80$	0	0
$81$	19.8611	2.20679
$82$	0	0
$83$	−11.0579	−1.21376	−0.606882	−	0.794792i	$-0.707580\pi$
$83$	−11.0579	−1.21376	−0.606882	+	0.794792i	$0.707580\pi$
$84$	0	0
$85$	7.71872	0.837212
$86$	0	0
$87$	1.32404	0.141952
$88$	0	0
$89$	3.91713	0.415215	0.207608	−	0.978212i	$-0.433432\pi$
$89$	3.91713	0.415215	0.207608	+	0.978212i	$0.433432\pi$
$90$	0	0
$91$	−17.7099	−1.85650
$92$	0	0
$93$	−20.4053	−2.11593
$94$	0	0
$95$	4.27638	0.438747
$96$	0	0
$97$	7.08577	0.719451	0.359726	−	0.933058i	$-0.382870\pi$
$97$	7.08577	0.719451	0.359726	+	0.933058i	$0.382870\pi$
$98$	0	0
$99$	25.7367	2.58664
$100$	0	0
$101$	6.28317	0.625199	0.312600	−	0.949885i	$-0.398800\pi$
$101$	6.28317	0.625199	0.312600	+	0.949885i	$0.398800\pi$
$102$	0	0
$103$	−1.36553	−0.134550	−0.0672750	−	0.997734i	$-0.521430\pi$
$103$	−1.36553	−0.134550	−0.0672750	+	0.997734i	$0.521430\pi$
$104$	0	0
$105$	64.2792	6.27301
$106$	0	0
$107$	−14.7230	−1.42333	−0.711664	−	0.702520i	$-0.752058\pi$
$107$	−14.7230	−1.42333	−0.711664	+	0.702520i	$0.752058\pi$
$108$	0	0
$109$	17.0448	1.63260	0.816299	−	0.577630i	$-0.196022\pi$
$109$	17.0448	1.63260	0.816299	+	0.577630i	$0.196022\pi$
$110$	0	0
$111$	−15.8000	−1.49967
$112$	0	0
$113$	−5.32408	−0.500847	−0.250424	−	0.968136i	$-0.580570\pi$
$113$	−5.32408	−0.500847	−0.250424	+	0.968136i	$0.580570\pi$
$114$	0	0
$115$	10.6322	0.991453
$116$	0	0
$117$	26.4726	2.44740
$118$	0	0
$119$	8.54636	0.783444
$120$	0	0
$121$	2.22264	0.202058
$122$	0	0
$123$	−0.0411964	−0.00371455
$124$	0	0
$125$	−35.4402	−3.16987
$126$	0	0
$127$	−11.2335	−0.996814	−0.498407	−	0.866943i	$-0.666081\pi$
$127$	−11.2335	−0.996814	−0.498407	+	0.866943i	$0.666081\pi$
$128$	0	0
$129$	−35.5154	−3.12696
$130$	0	0
$131$	−2.46946	−0.215758	−0.107879	−	0.994164i	$-0.534406\pi$
$131$	−2.46946	−0.215758	−0.107879	+	0.994164i	$0.534406\pi$
$132$	0	0
$133$	4.73492	0.410570
$134$	0	0
$135$	−55.3575	−4.76442
$136$	0	0
$137$	8.84634	0.755794	0.377897	−	0.925848i	$-0.376647\pi$
$137$	8.84634	0.755794	0.377897	+	0.925848i	$0.376647\pi$
$138$	0	0
$139$	15.0915	1.28004	0.640021	−	0.768358i	$-0.278926\pi$
$139$	15.0915	1.28004	0.640021	+	0.768358i	$0.278926\pi$
$140$	0	0
$141$	−19.4212	−1.63556
$142$	0	0
$143$	13.6007	1.13735
$144$	0	0
$145$	−1.78360	−0.148120
$146$	0	0
$147$	48.9497	4.03731
$148$	0	0
$149$	3.63193	0.297540	0.148770	−	0.988872i	$-0.452469\pi$
$149$	3.63193	0.297540	0.148770	+	0.988872i	$0.452469\pi$
$150$	0	0
$151$	5.70641	0.464381	0.232190	−	0.972670i	$-0.425411\pi$
$151$	5.70641	0.464381	0.232190	+	0.972670i	$0.425411\pi$
$152$	0	0
$153$	−12.7751	−1.03280
$154$	0	0
$155$	27.4877	2.20786
$156$	0	0
$157$	−16.8079	−1.34142	−0.670709	−	0.741721i	$-0.734010\pi$
$157$	−16.8079	−1.34142	−0.670709	+	0.741721i	$0.734010\pi$
$158$	0	0
$159$	3.17454	0.251758
$160$	0	0
$161$	11.7722	0.927779
$162$	0	0
$163$	−22.2051	−1.73924	−0.869619	−	0.493724i	$-0.835635\pi$
$163$	−22.2051	−1.73924	−0.869619	+	0.493724i	$0.835635\pi$
$164$	0	0
$165$	−49.3647	−3.84304
$166$	0	0
$167$	10.4987	0.812417	0.406208	−	0.913780i	$-0.366851\pi$
$167$	10.4987	0.812417	0.406208	+	0.913780i	$0.366851\pi$
$168$	0	0
$169$	0.989635	0.0761258
$170$	0	0
$171$	−7.07773	−0.541248
$172$	0	0
$173$	−13.0826	−0.994649	−0.497324	−	0.867565i	$-0.665684\pi$
$173$	−13.0826	−0.994649	−0.497324	+	0.867565i	$0.665684\pi$
$174$	0	0
$175$	−62.9149	−4.75592
$176$	0	0
$177$	−39.6057	−2.97695
$178$	0	0
$179$	18.7498	1.40142	0.700712	−	0.713444i	$-0.252866\pi$
$179$	18.7498	1.40142	0.700712	+	0.713444i	$0.252866\pi$
$180$	0	0
$181$	17.9054	1.33090	0.665449	−	0.746443i	$-0.268240\pi$
$181$	17.9054	1.33090	0.665449	+	0.746443i	$0.268240\pi$
$182$	0	0
$183$	13.9884	1.03405
$184$	0	0
$185$	21.2840	1.56483
$186$	0	0
$187$	−6.56338	−0.479962
$188$	0	0
$189$	−61.2933	−4.45843
$190$	0	0
$191$	−24.2695	−1.75608	−0.878039	−	0.478588i	$-0.841149\pi$
$191$	−24.2695	−1.75608	−0.878039	+	0.478588i	$0.841149\pi$
$192$	0	0
$193$	−27.6589	−1.99093	−0.995467	−	0.0951104i	$-0.969680\pi$
$193$	−27.6589	−1.99093	−0.995467	+	0.0951104i	$0.969680\pi$
$194$	0	0
$195$	−50.7763	−3.63617
$196$	0	0
$197$	17.5481	1.25025	0.625126	−	0.780524i	$-0.285047\pi$
$197$	17.5481	1.25025	0.625126	+	0.780524i	$0.285047\pi$
$198$	0	0
$199$	−7.03326	−0.498575	−0.249287	−	0.968430i	$-0.580196\pi$
$199$	−7.03326	−0.498575	−0.249287	+	0.968430i	$0.580196\pi$
$200$	0	0
$201$	−45.1859	−3.18717
$202$	0	0
$203$	−1.97484	−0.138607
$204$	0	0
$205$	0.0554950	0.00387594
$206$	0	0
$207$	−17.5970	−1.22308
$208$	0	0
$209$	−3.63629	−0.251528
$210$	0	0
$211$	−9.66387	−0.665288	−0.332644	−	0.943052i	$-0.607941\pi$
$211$	−9.66387	−0.665288	−0.332644	+	0.943052i	$0.607941\pi$
$212$	0	0
$213$	13.4708	0.923005
$214$	0	0
$215$	47.8423	3.26282
$216$	0	0
$217$	30.4351	2.06607
$218$	0	0
$219$	−34.2796	−2.31640
$220$	0	0
$221$	−6.75106	−0.454125
$222$	0	0
$223$	12.1280	0.812151	0.406075	−	0.913840i	$-0.366897\pi$
$223$	12.1280	0.812151	0.406075	+	0.913840i	$0.366897\pi$
$224$	0	0
$225$	94.0449	6.26966
$226$	0	0
$227$	18.5331	1.23009	0.615043	−	0.788493i	$-0.289139\pi$
$227$	18.5331	1.23009	0.615043	+	0.788493i	$0.289139\pi$
$228$	0	0
$229$	−21.6898	−1.43330	−0.716650	−	0.697433i	$-0.754326\pi$
$229$	−21.6898	−1.43330	−0.716650	+	0.697433i	$0.754326\pi$
$230$	0	0
$231$	−54.6579	−3.59623
$232$	0	0
$233$	−6.77239	−0.443674	−0.221837	−	0.975084i	$-0.571205\pi$
$233$	−6.77239	−0.443674	−0.221837	+	0.975084i	$0.571205\pi$
$234$	0	0
$235$	26.1620	1.70662
$236$	0	0
$237$	−12.4865	−0.811086
$238$	0	0
$239$	7.16525	0.463481	0.231741	−	0.972778i	$-0.425558\pi$
$239$	7.16525	0.463481	0.231741	+	0.972778i	$0.425558\pi$
$240$	0	0
$241$	−2.06497	−0.133017	−0.0665083	−	0.997786i	$-0.521186\pi$
$241$	−2.06497	−0.133017	−0.0665083	+	0.997786i	$0.521186\pi$
$242$	0	0
$243$	24.2151	1.55340
$244$	0	0
$245$	−65.9395	−4.21272
$246$	0	0
$247$	−3.74027	−0.237988
$248$	0	0
$249$	−35.1038	−2.22462
$250$	0	0
$251$	7.87980	0.497369	0.248684	−	0.968585i	$-0.420002\pi$
$251$	7.87980	0.497369	0.248684	+	0.968585i	$0.420002\pi$
$252$	0	0
$253$	−9.04074	−0.568386
$254$	0	0
$255$	24.5034	1.53446
$256$	0	0
$257$	24.1450	1.50613	0.753063	−	0.657948i	$-0.228575\pi$
$257$	24.1450	1.50613	0.753063	+	0.657948i	$0.228575\pi$
$258$	0	0
$259$	23.5662	1.46433
$260$	0	0
$261$	2.95199	0.182723
$262$	0	0
$263$	−9.62011	−0.593202	−0.296601	−	0.955002i	$-0.595853\pi$
$263$	−9.62011	−0.593202	−0.296601	+	0.955002i	$0.595853\pi$
$264$	0	0
$265$	−4.27638	−0.262696
$266$	0	0
$267$	12.4351	0.761017
$268$	0	0
$269$	20.6103	1.25663	0.628316	−	0.777958i	$-0.283745\pi$
$269$	20.6103	1.25663	0.628316	+	0.777958i	$0.283745\pi$
$270$	0	0
$271$	−14.7013	−0.893040	−0.446520	−	0.894774i	$-0.647337\pi$
$271$	−14.7013	−0.893040	−0.446520	+	0.894774i	$0.647337\pi$
$272$	0	0
$273$	−56.2208	−3.40264
$274$	0	0
$275$	48.3170	2.91363
$276$	0	0
$277$	3.43234	0.206229	0.103115	−	0.994669i	$-0.467119\pi$
$277$	3.43234	0.206229	0.103115	+	0.994669i	$0.467119\pi$
$278$	0	0
$279$	−45.4942	−2.72367
$280$	0	0
$281$	−28.8272	−1.71968	−0.859842	−	0.510560i	$-0.829438\pi$
$281$	−28.8272	−1.71968	−0.859842	+	0.510560i	$0.829438\pi$
$282$	0	0
$283$	31.7392	1.88670	0.943349	−	0.331802i	$-0.107657\pi$
$283$	31.7392	1.88670	0.943349	+	0.331802i	$0.107657\pi$
$284$	0	0
$285$	13.5756	0.804147
$286$	0	0
$287$	0.0614455	0.00362701
$288$	0	0
$289$	−13.7421	−0.808359
$290$	0	0
$291$	22.4941	1.31863
$292$	0	0
$293$	−8.61913	−0.503535	−0.251767	−	0.967788i	$-0.581012\pi$
$293$	−8.61913	−0.503535	−0.251767	+	0.967788i	$0.581012\pi$
$294$	0	0
$295$	53.3522	3.10629
$296$	0	0
$297$	47.0716	2.73137
$298$	0	0
$299$	−9.29925	−0.537790
$300$	0	0
$301$	52.9722	3.05327
$302$	0	0
$303$	19.9462	1.14588
$304$	0	0
$305$	−18.8436	−1.07898
$306$	0	0
$307$	16.1844	0.923694	0.461847	−	0.886960i	$-0.347187\pi$
$307$	16.1844	0.923694	0.461847	+	0.886960i	$0.347187\pi$
$308$	0	0
$309$	−4.33495	−0.246607
$310$	0	0
$311$	−4.62784	−0.262421	−0.131210	−	0.991355i	$-0.541886\pi$
$311$	−4.62784	−0.262421	−0.131210	+	0.991355i	$0.541886\pi$
$312$	0	0
$313$	−7.99505	−0.451907	−0.225953	−	0.974138i	$-0.572550\pi$
$313$	−7.99505	−0.451907	−0.225953	+	0.974138i	$0.572550\pi$
$314$	0	0
$315$	143.312	8.07473
$316$	0	0
$317$	23.0306	1.29353	0.646763	−	0.762691i	$-0.276122\pi$
$317$	23.0306	1.29353	0.646763	+	0.762691i	$0.276122\pi$
$318$	0	0
$319$	1.51663	0.0849149
$320$	0	0
$321$	−46.7389	−2.60871
$322$	0	0
$323$	1.80496	0.100431
$324$	0	0
$325$	49.6986	2.75678
$326$	0	0
$327$	54.1095	2.99226
$328$	0	0
$329$	28.9672	1.59702
$330$	0	0
$331$	9.06911	0.498484	0.249242	−	0.968441i	$-0.419819\pi$
$331$	9.06911	0.498484	0.249242	+	0.968441i	$0.419819\pi$
$332$	0	0
$333$	−35.2266	−1.93041
$334$	0	0
$335$	60.8693	3.32564
$336$	0	0
$337$	−11.6223	−0.633108	−0.316554	−	0.948575i	$-0.602526\pi$
$337$	−11.6223	−0.633108	−0.316554	+	0.948575i	$0.602526\pi$
$338$	0	0
$339$	−16.9015	−0.917965
$340$	0	0
$341$	−23.3733	−1.26574
$342$	0	0
$343$	−39.8654	−2.15253
$344$	0	0
$345$	33.7522	1.81716
$346$	0	0
$347$	9.64973	0.518024	0.259012	−	0.965874i	$-0.416603\pi$
$347$	9.64973	0.518024	0.259012	+	0.965874i	$0.416603\pi$
$348$	0	0
$349$	12.8659	0.688695	0.344347	−	0.938842i	$-0.388100\pi$
$349$	12.8659	0.688695	0.344347	+	0.938842i	$0.388100\pi$
$350$	0	0
$351$	48.4176	2.58434
$352$	0	0
$353$	−25.3810	−1.35089	−0.675446	−	0.737410i	$-0.736049\pi$
$353$	−25.3810	−1.35089	−0.675446	+	0.737410i	$0.736049\pi$
$354$	0	0
$355$	−18.1463	−0.963108
$356$	0	0
$357$	27.1308	1.43591
$358$	0	0
$359$	−8.73670	−0.461106	−0.230553	−	0.973060i	$-0.574053\pi$
$359$	−8.73670	−0.461106	−0.230553	+	0.973060i	$0.574053\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	7.05587	0.370337
$364$	0	0
$365$	46.1775	2.41704
$366$	0	0
$367$	10.8751	0.567675	0.283838	−	0.958872i	$-0.408392\pi$
$367$	10.8751	0.567675	0.283838	+	0.958872i	$0.408392\pi$
$368$	0	0
$369$	−0.0918484	−0.00478144
$370$	0	0
$371$	−4.73492	−0.245825
$372$	0	0
$373$	−0.610842	−0.0316282	−0.0158141	−	0.999875i	$-0.505034\pi$
$373$	−0.610842	−0.0316282	−0.0158141	+	0.999875i	$0.505034\pi$
$374$	0	0
$375$	−112.507	−5.80981
$376$	0	0
$377$	1.56000	0.0803438
$378$	0	0
$379$	−1.22331	−0.0628374	−0.0314187	−	0.999506i	$-0.510003\pi$
$379$	−1.22331	−0.0628374	−0.0314187	+	0.999506i	$0.510003\pi$
$380$	0	0
$381$	−35.6613	−1.82698
$382$	0	0
$383$	−4.17965	−0.213570	−0.106785	−	0.994282i	$-0.534056\pi$
$383$	−4.17965	−0.213570	−0.106785	+	0.994282i	$0.534056\pi$
$384$	0	0
$385$	73.6288	3.75247
$386$	0	0
$387$	−79.1826	−4.02508
$388$	0	0
$389$	11.6166	0.588984	0.294492	−	0.955654i	$-0.404850\pi$
$389$	11.6166	0.588984	0.294492	+	0.955654i	$0.404850\pi$
$390$	0	0
$391$	4.48759	0.226947
$392$	0	0
$393$	−7.83941	−0.395446
$394$	0	0
$395$	16.8204	0.846325
$396$	0	0
$397$	−17.0512	−0.855776	−0.427888	−	0.903832i	$-0.640742\pi$
$397$	−17.0512	−0.855776	−0.427888	+	0.903832i	$0.640742\pi$
$398$	0	0
$399$	15.0312	0.752502
$400$	0	0
$401$	7.21698	0.360399	0.180199	−	0.983630i	$-0.442326\pi$
$401$	7.21698	0.360399	0.180199	+	0.983630i	$0.442326\pi$
$402$	0	0
$403$	−24.0417	−1.19760
$404$	0	0
$405$	−84.9337	−4.22039
$406$	0	0
$407$	−18.0982	−0.897096
$408$	0	0
$409$	31.7986	1.57234	0.786170	−	0.618010i	$-0.212061\pi$
$409$	31.7986	1.57234	0.786170	+	0.618010i	$0.212061\pi$
$410$	0	0
$411$	28.0831	1.38524
$412$	0	0
$413$	59.0729	2.90679
$414$	0	0
$415$	47.2879	2.32127
$416$	0	0
$417$	47.9085	2.34609
$418$	0	0
$419$	28.4319	1.38899	0.694494	−	0.719499i	$-0.255628\pi$
$419$	28.4319	1.38899	0.694494	+	0.719499i	$0.255628\pi$
$420$	0	0
$421$	−22.8992	−1.11604	−0.558019	−	0.829828i	$-0.688438\pi$
$421$	−22.8992	−1.11604	−0.558019	+	0.829828i	$0.688438\pi$
$422$	0	0
$423$	−43.3001	−2.10532
$424$	0	0
$425$	−23.9834	−1.16336
$426$	0	0
$427$	−20.8641	−1.00968
$428$	0	0
$429$	43.1761	2.08456
$430$	0	0
$431$	12.3161	0.593247	0.296623	−	0.954995i	$-0.404139\pi$
$431$	12.3161	0.593247	0.296623	+	0.954995i	$0.404139\pi$
$432$	0	0
$433$	−0.236191	−0.0113506	−0.00567530	−	0.999984i	$-0.501807\pi$
$433$	−0.236191	−0.0113506	−0.00567530	+	0.999984i	$0.501807\pi$
$434$	0	0
$435$	−5.66210	−0.271477
$436$	0	0
$437$	2.48625	0.118933
$438$	0	0
$439$	−4.88355	−0.233079	−0.116540	−	0.993186i	$-0.537180\pi$
$439$	−4.88355	−0.233079	−0.116540	+	0.993186i	$0.537180\pi$
$440$	0	0
$441$	109.135	5.19689
$442$	0	0
$443$	26.2317	1.24630	0.623152	−	0.782101i	$-0.285852\pi$
$443$	26.2317	1.24630	0.623152	+	0.782101i	$0.285852\pi$
$444$	0	0
$445$	−16.7512	−0.794081
$446$	0	0
$447$	11.5297	0.545338
$448$	0	0
$449$	2.34028	0.110445	0.0552224	−	0.998474i	$-0.482413\pi$
$449$	2.34028	0.110445	0.0552224	+	0.998474i	$0.482413\pi$
$450$	0	0
$451$	−0.0471885	−0.00222202
$452$	0	0
$453$	18.1152	0.851128
$454$	0	0
$455$	75.7342	3.55047
$456$	0	0
$457$	8.04674	0.376410	0.188205	−	0.982130i	$-0.439733\pi$
$457$	8.04674	0.376410	0.188205	+	0.982130i	$0.439733\pi$
$458$	0	0
$459$	−23.3652	−1.09059
$460$	0	0
$461$	37.6901	1.75540	0.877702	−	0.479206i	$-0.159075\pi$
$461$	37.6901	1.75540	0.877702	+	0.479206i	$0.159075\pi$
$462$	0	0
$463$	5.57700	0.259185	0.129592	−	0.991567i	$-0.458633\pi$
$463$	5.57700	0.259185	0.129592	+	0.991567i	$0.458633\pi$
$464$	0	0
$465$	87.2609	4.04663
$466$	0	0
$467$	−30.5446	−1.41343	−0.706717	−	0.707496i	$-0.749825\pi$
$467$	−30.5446	−1.41343	−0.706717	+	0.707496i	$0.749825\pi$
$468$	0	0
$469$	67.3960	3.11206
$470$	0	0
$471$	−53.3575	−2.45858
$472$	0	0
$473$	−40.6813	−1.87053
$474$	0	0
$475$	−13.2874	−0.609669
$476$	0	0
$477$	7.07773	0.324067
$478$	0	0
$479$	1.63999	0.0749330	0.0374665	−	0.999298i	$-0.488071\pi$
$479$	1.63999	0.0749330	0.0374665	+	0.999298i	$0.488071\pi$
$480$	0	0
$481$	−18.6157	−0.848804
$482$	0	0
$483$	37.3714	1.70046
$484$	0	0
$485$	−30.3015	−1.37592
$486$	0	0
$487$	−10.3716	−0.469983	−0.234991	−	0.971997i	$-0.575506\pi$
$487$	−10.3716	−0.469983	−0.234991	+	0.971997i	$0.575506\pi$
$488$	0	0
$489$	−70.4911	−3.18772
$490$	0	0
$491$	−38.0504	−1.71719	−0.858595	−	0.512654i	$-0.828662\pi$
$491$	−38.0504	−1.71719	−0.858595	+	0.512654i	$0.828662\pi$
$492$	0	0
$493$	−0.752816	−0.0339051
$494$	0	0
$495$	−110.060	−4.94683
$496$	0	0
$497$	−20.0921	−0.901253
$498$	0	0
$499$	−33.9758	−1.52097	−0.760483	−	0.649358i	$-0.775038\pi$
$499$	−33.9758	−1.52097	−0.760483	+	0.649358i	$0.775038\pi$
$500$	0	0
$501$	33.3287	1.48902
$502$	0	0
$503$	13.4820	0.601132	0.300566	−	0.953761i	$-0.402824\pi$
$503$	13.4820	0.601132	0.300566	+	0.953761i	$0.402824\pi$
$504$	0	0
$505$	−26.8692	−1.19567
$506$	0	0
$507$	3.14164	0.139525
$508$	0	0
$509$	13.3229	0.590529	0.295264	−	0.955416i	$-0.404592\pi$
$509$	13.3229	0.590529	0.295264	+	0.955416i	$0.404592\pi$
$510$	0	0
$511$	51.1289	2.26181
$512$	0	0
$513$	−12.9449	−0.571533
$514$	0	0
$515$	5.83954	0.257321
$516$	0	0
$517$	−22.2461	−0.978382
$518$	0	0
$519$	−41.5312	−1.82302
$520$	0	0
$521$	−6.51367	−0.285369	−0.142685	−	0.989768i	$-0.545573\pi$
$521$	−6.51367	−0.285369	−0.142685	+	0.989768i	$0.545573\pi$
$522$	0	0
$523$	−9.38391	−0.410330	−0.205165	−	0.978727i	$-0.565773\pi$
$523$	−9.38391	−0.410330	−0.205165	+	0.978727i	$0.565773\pi$
$524$	0	0
$525$	−199.726	−8.71677
$526$	0	0
$527$	11.6019	0.505388
$528$	0	0
$529$	−16.8186	−0.731242
$530$	0	0
$531$	−88.3019	−3.83198
$532$	0	0
$533$	−0.0485379	−0.00210241
$534$	0	0
$535$	62.9612	2.72205
$536$	0	0
$537$	59.5220	2.56856
$538$	0	0
$539$	56.0697	2.41509
$540$	0	0
$541$	−2.62461	−0.112841	−0.0564203	−	0.998407i	$-0.517969\pi$
$541$	−2.62461	−0.112841	−0.0564203	+	0.998407i	$0.517969\pi$
$542$	0	0
$543$	56.8415	2.43930
$544$	0	0
$545$	−72.8901	−3.12227
$546$	0	0
$547$	−1.94741	−0.0832651	−0.0416325	−	0.999133i	$-0.513256\pi$
$547$	−1.94741	−0.0832651	−0.0416325	+	0.999133i	$0.513256\pi$
$548$	0	0
$549$	31.1876	1.33105
$550$	0	0
$551$	−0.417081	−0.0177682
$552$	0	0
$553$	18.6240	0.791971
$554$	0	0
$555$	67.5670	2.86806
$556$	0	0
$557$	20.9226	0.886518	0.443259	−	0.896393i	$-0.353822\pi$
$557$	20.9226	0.886518	0.443259	+	0.896393i	$0.353822\pi$
$558$	0	0
$559$	−41.8445	−1.76984
$560$	0	0
$561$	−20.8358	−0.879686
$562$	0	0
$563$	−19.8069	−0.834762	−0.417381	−	0.908732i	$-0.637052\pi$
$563$	−19.8069	−0.834762	−0.417381	+	0.908732i	$0.637052\pi$
$564$	0	0
$565$	22.7678	0.957848
$566$	0	0
$567$	−94.0407	−3.94934
$568$	0	0
$569$	9.47854	0.397361	0.198680	−	0.980064i	$-0.436334\pi$
$569$	9.47854	0.397361	0.198680	+	0.980064i	$0.436334\pi$
$570$	0	0
$571$	20.5826	0.861356	0.430678	−	0.902506i	$-0.358275\pi$
$571$	20.5826	0.861356	0.430678	+	0.902506i	$0.358275\pi$
$572$	0	0
$573$	−77.0446	−3.21858
$574$	0	0
$575$	−33.0359	−1.37769
$576$	0	0
$577$	3.18773	0.132707	0.0663534	−	0.997796i	$-0.478864\pi$
$577$	3.18773	0.132707	0.0663534	+	0.997796i	$0.478864\pi$
$578$	0	0
$579$	−87.8045	−3.64903
$580$	0	0
$581$	52.3583	2.17219
$582$	0	0
$583$	3.63629	0.150600
$584$	0	0
$585$	−113.207	−4.68054
$586$	0	0
$587$	40.7227	1.68081	0.840403	−	0.541962i	$-0.182318\pi$
$587$	40.7227	1.68081	0.840403	+	0.541962i	$0.182318\pi$
$588$	0	0
$589$	6.42779	0.264852
$590$	0	0
$591$	55.7073	2.29149
$592$	0	0
$593$	−0.612310	−0.0251446	−0.0125723	−	0.999921i	$-0.504002\pi$
$593$	−0.612310	−0.0251446	−0.0125723	+	0.999921i	$0.504002\pi$
$594$	0	0
$595$	−36.5475	−1.49830
$596$	0	0
$597$	−22.3274	−0.913800
$598$	0	0
$599$	−29.7994	−1.21757	−0.608785	−	0.793335i	$-0.708343\pi$
$599$	−29.7994	−1.21757	−0.608785	+	0.793335i	$0.708343\pi$
$600$	0	0
$601$	13.3764	0.545635	0.272818	−	0.962066i	$-0.412044\pi$
$601$	13.3764	0.545635	0.272818	+	0.962066i	$0.412044\pi$
$602$	0	0
$603$	−100.743	−4.10258
$604$	0	0
$605$	−9.50485	−0.386427
$606$	0	0
$607$	−24.9370	−1.01216	−0.506080	−	0.862486i	$-0.668906\pi$
$607$	−24.9370	−1.01216	−0.506080	+	0.862486i	$0.668906\pi$
$608$	0	0
$609$	−6.26923	−0.254042
$610$	0	0
$611$	−22.8822	−0.925715
$612$	0	0
$613$	7.02600	0.283778	0.141889	−	0.989883i	$-0.454682\pi$
$613$	7.02600	0.283778	0.141889	+	0.989883i	$0.454682\pi$
$614$	0	0
$615$	0.176171	0.00710391
$616$	0	0
$617$	36.7738	1.48046	0.740229	−	0.672355i	$-0.234717\pi$
$617$	36.7738	1.48046	0.740229	+	0.672355i	$0.234717\pi$
$618$	0	0
$619$	−17.6301	−0.708613	−0.354306	−	0.935129i	$-0.615283\pi$
$619$	−17.6301	−0.708613	−0.354306	+	0.935129i	$0.615283\pi$
$620$	0	0
$621$	−32.1844	−1.29151
$622$	0	0
$623$	−18.5473	−0.743082
$624$	0	0
$625$	85.1187	3.40475
$626$	0	0
$627$	−11.5436	−0.461006
$628$	0	0
$629$	8.98350	0.358196
$630$	0	0
$631$	32.4321	1.29110	0.645551	−	0.763717i	$-0.276628\pi$
$631$	32.4321	1.29110	0.645551	+	0.763717i	$0.276628\pi$
$632$	0	0
$633$	−30.6784	−1.21936
$634$	0	0
$635$	48.0388	1.90636
$636$	0	0
$637$	57.6729	2.28509
$638$	0	0
$639$	30.0336	1.18811
$640$	0	0
$641$	−33.5398	−1.32474	−0.662371	−	0.749176i	$-0.730450\pi$
$641$	−33.5398	−1.32474	−0.662371	+	0.749176i	$0.730450\pi$
$642$	0	0
$643$	36.7000	1.44731	0.723654	−	0.690163i	$-0.242461\pi$
$643$	36.7000	1.44731	0.723654	+	0.690163i	$0.242461\pi$
$644$	0	0
$645$	151.877	5.98017
$646$	0	0
$647$	18.3697	0.722186	0.361093	−	0.932530i	$-0.382404\pi$
$647$	18.3697	0.722186	0.361093	+	0.932530i	$0.382404\pi$
$648$	0	0
$649$	−45.3665	−1.78079
$650$	0	0
$651$	96.6175	3.78674
$652$	0	0
$653$	−43.9517	−1.71996	−0.859982	−	0.510324i	$-0.829526\pi$
$653$	−43.9517	−1.71996	−0.859982	+	0.510324i	$0.829526\pi$
$654$	0	0
$655$	10.5604	0.412627
$656$	0	0
$657$	−76.4273	−2.98171
$658$	0	0
$659$	15.3575	0.598245	0.299122	−	0.954215i	$-0.403306\pi$
$659$	15.3575	0.598245	0.299122	+	0.954215i	$0.403306\pi$
$660$	0	0
$661$	−37.3749	−1.45372	−0.726858	−	0.686787i	$-0.759020\pi$
$661$	−37.3749	−1.45372	−0.726858	+	0.686787i	$0.759020\pi$
$662$	0	0
$663$	−21.4315	−0.832332
$664$	0	0
$665$	−20.2483	−0.785196
$666$	0	0
$667$	−1.03697	−0.0401515
$668$	0	0
$669$	38.5009	1.48853
$670$	0	0
$671$	16.0231	0.618565
$672$	0	0
$673$	1.34765	0.0519481	0.0259741	−	0.999663i	$-0.491731\pi$
$673$	1.34765	0.0519481	0.0259741	+	0.999663i	$0.491731\pi$
$674$	0	0
$675$	172.005	6.62048
$676$	0	0
$677$	42.4717	1.63232	0.816160	−	0.577825i	$-0.196099\pi$
$677$	42.4717	1.63232	0.816160	+	0.577825i	$0.196099\pi$
$678$	0	0
$679$	−33.5505	−1.28755
$680$	0	0
$681$	58.8342	2.25453
$682$	0	0
$683$	−16.6167	−0.635820	−0.317910	−	0.948121i	$-0.602981\pi$
$683$	−16.6167	−0.635820	−0.317910	+	0.948121i	$0.602981\pi$
$684$	0	0
$685$	−37.8303	−1.44542
$686$	0	0
$687$	−68.8552	−2.62699
$688$	0	0
$689$	3.74027	0.142493
$690$	0	0
$691$	36.2051	1.37731	0.688654	−	0.725090i	$-0.258202\pi$
$691$	36.2051	1.37731	0.688654	+	0.725090i	$0.258202\pi$
$692$	0	0
$693$	−121.861	−4.62913
$694$	0	0
$695$	−64.5368	−2.44802
$696$	0	0
$697$	0.0234232	0.000887217 0
$698$	0	0
$699$	−21.4993	−0.813177
$700$	0	0
$701$	8.01274	0.302637	0.151319	−	0.988485i	$-0.451648\pi$
$701$	8.01274	0.302637	0.151319	+	0.988485i	$0.451648\pi$
$702$	0	0
$703$	4.97711	0.187715
$704$	0	0
$705$	83.0525	3.12794
$706$	0	0
$707$	−29.7503	−1.11888
$708$	0	0
$709$	−26.0454	−0.978155	−0.489077	−	0.872240i	$-0.662666\pi$
$709$	−26.0454	−0.978155	−0.489077	+	0.872240i	$0.662666\pi$
$710$	0	0
$711$	−27.8390	−1.04404
$712$	0	0
$713$	15.9811	0.598497
$714$	0	0
$715$	−58.1619	−2.17513
$716$	0	0
$717$	22.7464	0.849480
$718$	0	0
$719$	24.8824	0.927959	0.463979	−	0.885846i	$-0.346421\pi$
$719$	24.8824	0.927959	0.463979	+	0.885846i	$0.346421\pi$
$720$	0	0
$721$	6.46569	0.240795
$722$	0	0
$723$	−6.55535	−0.243796
$724$	0	0
$725$	5.54193	0.205822
$726$	0	0
$727$	7.24276	0.268619	0.134309	−	0.990939i	$-0.457118\pi$
$727$	7.24276	0.268619	0.134309	+	0.990939i	$0.457118\pi$
$728$	0	0
$729$	17.2887	0.640322
$730$	0	0
$731$	20.1932	0.746871
$732$	0	0
$733$	1.98752	0.0734107	0.0367053	−	0.999326i	$-0.488314\pi$
$733$	1.98752	0.0734107	0.0367053	+	0.999326i	$0.488314\pi$
$734$	0	0
$735$	−209.328	−7.72117
$736$	0	0
$737$	−51.7584	−1.90655
$738$	0	0
$739$	−29.7505	−1.09439	−0.547195	−	0.837005i	$-0.684305\pi$
$739$	−29.7505	−1.09439	−0.547195	+	0.837005i	$0.684305\pi$
$740$	0	0
$741$	−11.8737	−0.436190
$742$	0	0
$743$	−18.9921	−0.696752	−0.348376	−	0.937355i	$-0.613267\pi$
$743$	−18.9921	−0.696752	−0.348376	+	0.937355i	$0.613267\pi$
$744$	0	0
$745$	−15.5315	−0.569031
$746$	0	0
$747$	−78.2650	−2.86357
$748$	0	0
$749$	69.7123	2.54723
$750$	0	0
$751$	37.1163	1.35439	0.677196	−	0.735802i	$-0.263195\pi$
$751$	37.1163	1.35439	0.677196	+	0.735802i	$0.263195\pi$
$752$	0	0
$753$	25.0148	0.911589
$754$	0	0
$755$	−24.4028	−0.888108
$756$	0	0
$757$	13.1729	0.478776	0.239388	−	0.970924i	$-0.423053\pi$
$757$	13.1729	0.478776	0.239388	+	0.970924i	$0.423053\pi$
$758$	0	0
$759$	−28.7002	−1.04175
$760$	0	0
$761$	54.5440	1.97722	0.988610	−	0.150503i	$-0.0480893\pi$
$761$	54.5440	1.97722	0.988610	+	0.150503i	$0.0480893\pi$
$762$	0	0
$763$	−80.7058	−2.92175
$764$	0	0
$765$	54.6310	1.97519
$766$	0	0
$767$	−46.6637	−1.68493
$768$	0	0
$769$	−33.6028	−1.21175	−0.605873	−	0.795561i	$-0.707176\pi$
$769$	−33.6028	−1.21175	−0.605873	+	0.795561i	$0.707176\pi$
$770$	0	0
$771$	76.6495	2.76046
$772$	0	0
$773$	−3.50565	−0.126090	−0.0630448	−	0.998011i	$-0.520081\pi$
$773$	−3.50565	−0.126090	−0.0630448	+	0.998011i	$0.520081\pi$
$774$	0	0
$775$	−85.4088	−3.06798
$776$	0	0
$777$	74.8119	2.68386
$778$	0	0
$779$	0.0129771	0.000464953 0
$780$	0	0
$781$	15.4302	0.552136
$782$	0	0
$783$	5.39909	0.192948
$784$	0	0
$785$	71.8771	2.56540
$786$	0	0
$787$	−20.7607	−0.740041	−0.370020	−	0.929024i	$-0.620649\pi$
$787$	−20.7607	−0.740041	−0.370020	+	0.929024i	$0.620649\pi$
$788$	0	0
$789$	−30.5395	−1.08723
$790$	0	0
$791$	25.2091	0.896332
$792$	0	0
$793$	16.4813	0.585267
$794$	0	0
$795$	−13.5756	−0.481476
$796$	0	0
$797$	−16.5479	−0.586156	−0.293078	−	0.956088i	$-0.594680\pi$
$797$	−16.5479	−0.586156	−0.293078	+	0.956088i	$0.594680\pi$
$798$	0	0
$799$	11.0424	0.390652
$800$	0	0
$801$	27.7244	0.979595
$802$	0	0
$803$	−39.2657	−1.38566
$804$	0	0
$805$	−50.3424	−1.77434
$806$	0	0
$807$	65.4283	2.30319
$808$	0	0
$809$	49.4756	1.73947	0.869735	−	0.493519i	$-0.164290\pi$
$809$	49.4756	1.73947	0.869735	+	0.493519i	$0.164290\pi$
$810$	0	0
$811$	−37.7298	−1.32487	−0.662436	−	0.749119i	$-0.730477\pi$
$811$	−37.7298	−1.32487	−0.662436	+	0.749119i	$0.730477\pi$
$812$	0	0
$813$	−46.6699	−1.63679
$814$	0	0
$815$	94.9574	3.32621
$816$	0	0
$817$	11.1876	0.391403
$818$	0	0
$819$	−125.346	−4.37994
$820$	0	0
$821$	3.71452	0.129638	0.0648188	−	0.997897i	$-0.479353\pi$
$821$	3.71452	0.129638	0.0648188	+	0.997897i	$0.479353\pi$
$822$	0	0
$823$	45.1834	1.57500	0.787498	−	0.616317i	$-0.211376\pi$
$823$	45.1834	1.57500	0.787498	+	0.616317i	$0.211376\pi$
$824$	0	0
$825$	153.385	5.34016
$826$	0	0
$827$	−11.7935	−0.410099	−0.205050	−	0.978752i	$-0.565736\pi$
$827$	−11.7935	−0.410099	−0.205050	+	0.978752i	$0.565736\pi$
$828$	0	0
$829$	−16.8515	−0.585276	−0.292638	−	0.956223i	$-0.594533\pi$
$829$	−16.8515	−0.585276	−0.292638	+	0.956223i	$0.594533\pi$
$830$	0	0
$831$	10.8961	0.377982
$832$	0	0
$833$	−27.8316	−0.964307
$834$	0	0
$835$	−44.8966	−1.55371
$836$	0	0
$837$	−83.2074	−2.87607
$838$	0	0
$839$	−4.25734	−0.146980	−0.0734898	−	0.997296i	$-0.523414\pi$
$839$	−4.25734	−0.146980	−0.0734898	+	0.997296i	$0.523414\pi$
$840$	0	0
$841$	−28.8260	−0.994002
$842$	0	0
$843$	−91.5131	−3.15188
$844$	0	0
$845$	−4.23206	−0.145587
$846$	0	0
$847$	−10.5240	−0.361609
$848$	0	0
$849$	100.757	3.45799
$850$	0	0
$851$	12.3743	0.424187
$852$	0	0
$853$	0.607038	0.0207846	0.0103923	−	0.999946i	$-0.496692\pi$
$853$	0.607038	0.0207846	0.0103923	+	0.999946i	$0.496692\pi$
$854$	0	0
$855$	30.2671	1.03511
$856$	0	0
$857$	0.617490	0.0210931	0.0105465	−	0.999944i	$-0.496643\pi$
$857$	0.617490	0.0210931	0.0105465	+	0.999944i	$0.496643\pi$
$858$	0	0
$859$	15.0494	0.513478	0.256739	−	0.966481i	$-0.417352\pi$
$859$	15.0494	0.513478	0.256739	+	0.966481i	$0.417352\pi$
$860$	0	0
$861$	0.195061	0.00664767
$862$	0	0
$863$	−11.0514	−0.376195	−0.188097	−	0.982150i	$-0.560232\pi$
$863$	−11.0514	−0.376195	−0.188097	+	0.982150i	$0.560232\pi$
$864$	0	0
$865$	55.9460	1.90222
$866$	0	0
$867$	−43.6249	−1.48158
$868$	0	0
$869$	−14.3027	−0.485187
$870$	0	0
$871$	−53.2384	−1.80391
$872$	0	0
$873$	50.1512	1.69736
$874$	0	0
$875$	167.807	5.67290
$876$	0	0
$877$	−27.4907	−0.928294	−0.464147	−	0.885758i	$-0.653639\pi$
$877$	−27.4907	−0.928294	−0.464147	+	0.885758i	$0.653639\pi$
$878$	0	0
$879$	−27.3618	−0.922891
$880$	0	0
$881$	−8.63410	−0.290890	−0.145445	−	0.989366i	$-0.546461\pi$
$881$	−8.63410	−0.290890	−0.145445	+	0.989366i	$0.546461\pi$
$882$	0	0
$883$	−33.3515	−1.12237	−0.561183	−	0.827691i	$-0.689654\pi$
$883$	−33.3515	−1.12237	−0.561183	+	0.827691i	$0.689654\pi$
$884$	0	0
$885$	169.369	5.69328
$886$	0	0
$887$	12.0868	0.405834	0.202917	−	0.979196i	$-0.434958\pi$
$887$	12.0868	0.405834	0.202917	+	0.979196i	$0.434958\pi$
$888$	0	0
$889$	53.1898	1.78393
$890$	0	0
$891$	72.2208	2.41949
$892$	0	0
$893$	6.11779	0.204724
$894$	0	0
$895$	−80.1812	−2.68016
$896$	0	0
$897$	−29.5209	−0.985674
$898$	0	0
$899$	−2.68091	−0.0894132
$900$	0	0
$901$	−1.80496	−0.0601321
$902$	0	0
$903$	168.163	5.59610
$904$	0	0
$905$	−76.5703	−2.54528
$906$	0	0
$907$	−10.4689	−0.347615	−0.173807	−	0.984780i	$-0.555607\pi$
$907$	−10.4689	−0.347615	−0.173807	+	0.984780i	$0.555607\pi$
$908$	0	0
$909$	44.4706	1.47500
$910$	0	0
$911$	35.2074	1.16647	0.583236	−	0.812303i	$-0.301786\pi$
$911$	35.2074	1.16647	0.583236	+	0.812303i	$0.301786\pi$
$912$	0	0
$913$	−40.2098	−1.33075
$914$	0	0
$915$	−59.8198	−1.97758
$916$	0	0
$917$	11.6927	0.386127
$918$	0	0
$919$	−11.4368	−0.377266	−0.188633	−	0.982048i	$-0.560406\pi$
$919$	−11.4368	−0.377266	−0.188633	+	0.982048i	$0.560406\pi$
$920$	0	0
$921$	51.3782	1.69297
$922$	0	0
$923$	15.8714	0.522414
$924$	0	0
$925$	−66.1330	−2.17444
$926$	0	0
$927$	−9.66489	−0.317436
$928$	0	0
$929$	−28.3131	−0.928921	−0.464461	−	0.885594i	$-0.653752\pi$
$929$	−28.3131	−0.928921	−0.464461	+	0.885594i	$0.653752\pi$
$930$	0	0
$931$	−15.4195	−0.505352
$932$	0	0
$933$	−14.6913	−0.480971
$934$	0	0
$935$	28.0675	0.917906
$936$	0	0
$937$	−8.47783	−0.276959	−0.138479	−	0.990365i	$-0.544221\pi$
$937$	−8.47783	−0.276959	−0.138479	+	0.990365i	$0.544221\pi$
$938$	0	0
$939$	−25.3806	−0.828266
$940$	0	0
$941$	17.6707	0.576049	0.288025	−	0.957623i	$-0.407001\pi$
$941$	17.6707	0.576049	0.288025	+	0.957623i	$0.407001\pi$
$942$	0	0
$943$	0.0322643	0.00105067
$944$	0	0
$945$	262.113	8.52655
$946$	0	0
$947$	−13.9249	−0.452500	−0.226250	−	0.974069i	$-0.572647\pi$
$947$	−13.9249	−0.452500	−0.226250	+	0.974069i	$0.572647\pi$
$948$	0	0
$949$	−40.3885	−1.31107
$950$	0	0
$951$	73.1116	2.37080
$952$	0	0
$953$	54.2582	1.75760	0.878798	−	0.477193i	$-0.158346\pi$
$953$	54.2582	1.75760	0.878798	+	0.477193i	$0.158346\pi$
$954$	0	0
$955$	103.786	3.35842
$956$	0	0
$957$	4.81460	0.155634
$958$	0	0
$959$	−41.8867	−1.35259
$960$	0	0
$961$	10.3165	0.332790
$962$	0	0
$963$	−104.206	−3.35798
$964$	0	0
$965$	118.280	3.80757
$966$	0	0
$967$	−55.5567	−1.78658	−0.893292	−	0.449476i	$-0.851611\pi$
$967$	−55.5567	−1.78658	−0.893292	+	0.449476i	$0.851611\pi$
$968$	0	0
$969$	5.72994	0.184072
$970$	0	0
$971$	−38.6461	−1.24021	−0.620106	−	0.784518i	$-0.712910\pi$
$971$	−38.6461	−1.24021	−0.620106	+	0.784518i	$0.712910\pi$
$972$	0	0
$973$	−71.4569	−2.29080
$974$	0	0
$975$	157.770	5.05270
$976$	0	0
$977$	39.0149	1.24820	0.624099	−	0.781345i	$-0.285466\pi$
$977$	39.0149	1.24820	0.624099	+	0.781345i	$0.285466\pi$
$978$	0	0
$979$	14.2439	0.455236
$980$	0	0
$981$	120.639	3.85170
$982$	0	0
$983$	13.0682	0.416811	0.208406	−	0.978042i	$-0.433173\pi$
$983$	13.0682	0.416811	0.208406	+	0.978042i	$0.433173\pi$
$984$	0	0
$985$	−75.0425	−2.39105
$986$	0	0
$987$	91.9578	2.92705
$988$	0	0
$989$	27.8151	0.884468
$990$	0	0
$991$	−42.5907	−1.35294	−0.676468	−	0.736472i	$-0.736490\pi$
$991$	−42.5907	−1.35294	−0.676468	+	0.736472i	$0.736490\pi$
$992$	0	0
$993$	28.7903	0.913633
$994$	0	0
$995$	30.0769	0.953502
$996$	0	0
$997$	34.2479	1.08464	0.542321	−	0.840171i	$-0.317546\pi$
$997$	34.2479	1.08464	0.542321	+	0.840171i	$0.317546\pi$
$998$	0	0
$999$	−64.4284	−2.03842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.19	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.19	✓	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)

\(f(q)\)	\(=\)	\(q+3.17454 q^{3} -4.27638 q^{5} -4.73492 q^{7} +7.07773 q^{9} +O(q^{10})\)	\(q+3.17454 q^{3} -4.27638 q^{5} -4.73492 q^{7} +7.07773 q^{9} +3.63629 q^{11} +3.74027 q^{13} -13.5756 q^{15} -1.80496 q^{17} -1.00000 q^{19} -15.0312 q^{21} -2.48625 q^{23} +13.2874 q^{25} +12.9449 q^{27} +0.417081 q^{29} -6.42779 q^{31} +11.5436 q^{33} +20.2483 q^{35} -4.97711 q^{37} +11.8737 q^{39} -0.0129771 q^{41} -11.1876 q^{43} -30.2671 q^{45} -6.11779 q^{47} +15.4195 q^{49} -5.72994 q^{51} +1.00000 q^{53} -15.5502 q^{55} -3.17454 q^{57} -12.4760 q^{59} +4.40643 q^{61} -33.5125 q^{63} -15.9948 q^{65} -14.2338 q^{67} -7.89271 q^{69} +4.24339 q^{71} -10.7983 q^{73} +42.1815 q^{75} -17.2176 q^{77} -3.93332 q^{79} +19.8611 q^{81} -11.0579 q^{83} +7.71872 q^{85} +1.32404 q^{87} +3.91713 q^{89} -17.7099 q^{91} -20.4053 q^{93} +4.27638 q^{95} +7.08577 q^{97} +25.7367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 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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