LMFDB - Embedded newform 8028.2.a.j.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, 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("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 38x^{3} - 46x^{2} - 36x + 9 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 38x^3 - 46x^2 - 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 892)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.80512$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.13650 q^{5} -1.96968 q^{7} +O(q^{10})$	$q+4.13650 q^{5} -1.96968 q^{7} +2.47938 q^{11} -4.26095 q^{13} +3.11132 q^{17} -0.960395 q^{19} +4.32574 q^{23} +12.1107 q^{25} +0.724125 q^{29} +9.48409 q^{31} -8.14757 q^{35} -10.3400 q^{37} +0.835446 q^{41} +5.60510 q^{43} +5.42708 q^{47} -3.12038 q^{49} +5.22884 q^{53} +10.2560 q^{55} -9.73990 q^{59} +2.76465 q^{61} -17.6254 q^{65} -5.86871 q^{67} -12.7387 q^{71} +4.83460 q^{73} -4.88358 q^{77} +5.83650 q^{79} +13.7751 q^{83} +12.8700 q^{85} +14.5514 q^{89} +8.39269 q^{91} -3.97268 q^{95} -2.44098 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{5} - q^{7}+O(q^{10}) $ 7 * q + 7 * q^5 - q^7	$ 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) $ 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

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	0
$3$	0	0
$4$	0	0
$5$	4.13650	1.84990	0.924950	−	0.380088i	$-0.124106\pi$
$5$	4.13650	1.84990	0.924950	+	0.380088i	$0.124106\pi$
$6$	0	0
$7$	−1.96968	−0.744467	−0.372234	−	0.928139i	$-0.621408\pi$
$7$	−1.96968	−0.744467	−0.372234	+	0.928139i	$0.621408\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.47938	0.747562	0.373781	−	0.927517i	$-0.378061\pi$
$11$	2.47938	0.747562	0.373781	+	0.927517i	$0.378061\pi$
$12$	0	0
$13$	−4.26095	−1.18177	−0.590887	−	0.806754i	$-0.701222\pi$
$13$	−4.26095	−1.18177	−0.590887	+	0.806754i	$0.701222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.11132	0.754606	0.377303	−	0.926090i	$-0.376851\pi$
$17$	3.11132	0.754606	0.377303	+	0.926090i	$0.376851\pi$
$18$	0	0
$19$	−0.960395	−0.220330	−0.110165	−	0.993913i	$-0.535138\pi$
$19$	−0.960395	−0.220330	−0.110165	+	0.993913i	$0.535138\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.32574	0.901979	0.450989	−	0.892529i	$-0.351071\pi$
$23$	4.32574	0.901979	0.450989	+	0.892529i	$0.351071\pi$
$24$	0	0
$25$	12.1107	2.42213
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.724125	0.134467	0.0672334	−	0.997737i	$-0.478583\pi$
$29$	0.724125	0.134467	0.0672334	+	0.997737i	$0.478583\pi$
$30$	0	0
$31$	9.48409	1.70339	0.851697	−	0.524035i	$-0.175574\pi$
$31$	9.48409	1.70339	0.851697	+	0.524035i	$0.175574\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.14757	−1.37719
$36$	0	0
$37$	−10.3400	−1.69989	−0.849943	−	0.526874i	$-0.823364\pi$
$37$	−10.3400	−1.69989	−0.849943	+	0.526874i	$0.823364\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.835446	0.130475	0.0652374	−	0.997870i	$-0.479220\pi$
$41$	0.835446	0.130475	0.0652374	+	0.997870i	$0.479220\pi$
$42$	0	0
$43$	5.60510	0.854770	0.427385	−	0.904070i	$-0.359435\pi$
$43$	5.60510	0.854770	0.427385	+	0.904070i	$0.359435\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.42708	0.791621	0.395811	−	0.918332i	$-0.370464\pi$
$47$	5.42708	0.791621	0.395811	+	0.918332i	$0.370464\pi$
$48$	0	0
$49$	−3.12038	−0.445768
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.22884	0.718236	0.359118	−	0.933292i	$-0.383078\pi$
$53$	5.22884	0.718236	0.359118	+	0.933292i	$0.383078\pi$
$54$	0	0
$55$	10.2560	1.38292
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.73990	−1.26803	−0.634014	−	0.773322i	$-0.718594\pi$
$59$	−9.73990	−1.26803	−0.634014	+	0.773322i	$0.718594\pi$
$60$	0	0
$61$	2.76465	0.353978	0.176989	−	0.984213i	$-0.443364\pi$
$61$	2.76465	0.353978	0.176989	+	0.984213i	$0.443364\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17.6254	−2.18617
$66$	0	0
$67$	−5.86871	−0.716977	−0.358488	−	0.933534i	$-0.616708\pi$
$67$	−5.86871	−0.716977	−0.358488	+	0.933534i	$0.616708\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.7387	−1.51181	−0.755903	−	0.654684i	$-0.772802\pi$
$71$	−12.7387	−1.51181	−0.755903	+	0.654684i	$0.772802\pi$
$72$	0	0
$73$	4.83460	0.565847	0.282923	−	0.959143i	$-0.408696\pi$
$73$	4.83460	0.565847	0.282923	+	0.959143i	$0.408696\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.88358	−0.556536
$78$	0	0
$79$	5.83650	0.656657	0.328329	−	0.944564i	$-0.393515\pi$
$79$	5.83650	0.656657	0.328329	+	0.944564i	$0.393515\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.7751	1.51202	0.756009	−	0.654561i	$-0.227147\pi$
$83$	13.7751	1.51202	0.756009	+	0.654561i	$0.227147\pi$
$84$	0	0
$85$	12.8700	1.39595
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.5514	1.54244	0.771222	−	0.636566i	$-0.219646\pi$
$89$	14.5514	1.54244	0.771222	+	0.636566i	$0.219646\pi$
$90$	0	0
$91$	8.39269	0.879793
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.97268	−0.407588
$96$	0	0
$97$	−2.44098	−0.247843	−0.123922	−	0.992292i	$-0.539547\pi$
$97$	−2.44098	−0.247843	−0.123922	+	0.992292i	$0.539547\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.8875	−1.58087	−0.790435	−	0.612546i	$-0.790145\pi$
$101$	−15.8875	−1.58087	−0.790435	+	0.612546i	$0.790145\pi$
$102$	0	0
$103$	5.15105	0.507548	0.253774	−	0.967263i	$-0.418328\pi$
$103$	5.15105	0.507548	0.253774	+	0.967263i	$0.418328\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0952	1.45931	0.729656	−	0.683814i	$-0.239680\pi$
$107$	15.0952	1.45931	0.729656	+	0.683814i	$0.239680\pi$
$108$	0	0
$109$	−5.53963	−0.530600	−0.265300	−	0.964166i	$-0.585471\pi$
$109$	−5.53963	−0.530600	−0.265300	+	0.964166i	$0.585471\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.0565	−1.88676	−0.943380	−	0.331714i	$-0.892373\pi$
$113$	−20.0565	−1.88676	−0.943380	+	0.331714i	$0.892373\pi$
$114$	0	0
$115$	17.8934	1.66857
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.12829	−0.561780
$120$	0	0
$121$	−4.85266	−0.441151
$122$	0	0
$123$	0	0
$124$	0	0
$125$	29.4133	2.63081
$126$	0	0
$127$	−5.14943	−0.456938	−0.228469	−	0.973551i	$-0.573372\pi$
$127$	−5.14943	−0.456938	−0.228469	+	0.973551i	$0.573372\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.72174	0.762022	0.381011	−	0.924570i	$-0.375576\pi$
$131$	8.72174	0.762022	0.381011	+	0.924570i	$0.375576\pi$
$132$	0	0
$133$	1.89167	0.164028
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.70551	0.145711	0.0728557	−	0.997342i	$-0.476789\pi$
$137$	1.70551	0.145711	0.0728557	+	0.997342i	$0.476789\pi$
$138$	0	0
$139$	22.6104	1.91779	0.958895	−	0.283761i	$-0.0915822\pi$
$139$	22.6104	1.91779	0.958895	+	0.283761i	$0.0915822\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−10.5645	−0.883450
$144$	0	0
$145$	2.99535	0.248750
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.4644	0.939198	0.469599	−	0.882880i	$-0.344398\pi$
$149$	11.4644	0.939198	0.469599	+	0.882880i	$0.344398\pi$
$150$	0	0
$151$	2.97196	0.241855	0.120927	−	0.992661i	$-0.461413\pi$
$151$	2.97196	0.241855	0.120927	+	0.992661i	$0.461413\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	39.2310	3.15111
$156$	0	0
$157$	6.47361	0.516651	0.258325	−	0.966058i	$-0.416829\pi$
$157$	6.47361	0.516651	0.258325	+	0.966058i	$0.416829\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.52030	−0.671494
$162$	0	0
$163$	−18.3390	−1.43642	−0.718209	−	0.695828i	$-0.755038\pi$
$163$	−18.3390	−1.43642	−0.718209	+	0.695828i	$0.755038\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.8912	1.07493	0.537466	−	0.843285i	$-0.319382\pi$
$167$	13.8912	1.07493	0.537466	+	0.843285i	$0.319382\pi$
$168$	0	0
$169$	5.15569	0.396592
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.9181	1.89449	0.947243	−	0.320516i	$-0.103856\pi$
$173$	24.9181	1.89449	0.947243	+	0.320516i	$0.103856\pi$
$174$	0	0
$175$	−23.8541	−1.80320
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.97596	0.222433	0.111217	−	0.993796i	$-0.464525\pi$
$179$	2.97596	0.222433	0.111217	+	0.993796i	$0.464525\pi$
$180$	0	0
$181$	20.6526	1.53510	0.767549	−	0.640990i	$-0.221476\pi$
$181$	20.6526	1.53510	0.767549	+	0.640990i	$0.221476\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−42.7715	−3.14462
$186$	0	0
$187$	7.71416	0.564115
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.65220	−0.191907	−0.0959534	−	0.995386i	$-0.530590\pi$
$191$	−2.65220	−0.191907	−0.0959534	+	0.995386i	$0.530590\pi$
$192$	0	0
$193$	8.64291	0.622130	0.311065	−	0.950389i	$-0.399314\pi$
$193$	8.64291	0.622130	0.311065	+	0.950389i	$0.399314\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.712373	−0.0507544	−0.0253772	−	0.999678i	$-0.508079\pi$
$197$	−0.712373	−0.0507544	−0.0253772	+	0.999678i	$0.508079\pi$
$198$	0	0
$199$	10.4503	0.740799	0.370399	−	0.928873i	$-0.379221\pi$
$199$	10.4503	0.740799	0.370399	+	0.928873i	$0.379221\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.42629	−0.100106
$204$	0	0
$205$	3.45583	0.241365
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.38119	−0.164710
$210$	0	0
$211$	−21.9273	−1.50954	−0.754769	−	0.655991i	$-0.772251\pi$
$211$	−21.9273	−1.50954	−0.754769	+	0.655991i	$0.772251\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	23.1855	1.58124
$216$	0	0
$217$	−18.6806	−1.26812
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.2572	−0.891775
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.3594	1.68316	0.841580	−	0.540132i	$-0.181626\pi$
$227$	25.3594	1.68316	0.841580	+	0.540132i	$0.181626\pi$
$228$	0	0
$229$	15.9096	1.05133	0.525667	−	0.850690i	$-0.323816\pi$
$229$	15.9096	1.05133	0.525667	+	0.850690i	$0.323816\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.05957	−0.331463	−0.165732	−	0.986171i	$-0.552999\pi$
$233$	−5.05957	−0.331463	−0.165732	+	0.986171i	$0.552999\pi$
$234$	0	0
$235$	22.4491	1.46442
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.19119	0.271106	0.135553	−	0.990770i	$-0.456719\pi$
$239$	4.19119	0.271106	0.135553	+	0.990770i	$0.456719\pi$
$240$	0	0
$241$	−19.6165	−1.26361	−0.631805	−	0.775127i	$-0.717686\pi$
$241$	−19.6165	−1.26361	−0.631805	+	0.775127i	$0.717686\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−12.9075	−0.824627
$246$	0	0
$247$	4.09219	0.260380
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.8195	−0.872282	−0.436141	−	0.899878i	$-0.643655\pi$
$251$	−13.8195	−0.872282	−0.436141	+	0.899878i	$0.643655\pi$
$252$	0	0
$253$	10.7252	0.674285
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.18776	−0.198847	−0.0994234	−	0.995045i	$-0.531700\pi$
$257$	−3.18776	−0.198847	−0.0994234	+	0.995045i	$0.531700\pi$
$258$	0	0
$259$	20.3665	1.26551
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.2099	1.12287	0.561435	−	0.827521i	$-0.310250\pi$
$263$	18.2099	1.12287	0.561435	+	0.827521i	$0.310250\pi$
$264$	0	0
$265$	21.6291	1.32867
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.8522	0.661668	0.330834	−	0.943689i	$-0.392670\pi$
$269$	10.8522	0.661668	0.330834	+	0.943689i	$0.392670\pi$
$270$	0	0
$271$	−31.9951	−1.94356	−0.971782	−	0.235881i	$-0.924203\pi$
$271$	−31.9951	−1.94356	−0.971782	+	0.235881i	$0.924203\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	30.0270	1.81070
$276$	0	0
$277$	3.64409	0.218952	0.109476	−	0.993989i	$-0.465083\pi$
$277$	3.64409	0.218952	0.109476	+	0.993989i	$0.465083\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−23.1645	−1.38188	−0.690939	−	0.722913i	$-0.742803\pi$
$281$	−23.1645	−1.38188	−0.690939	+	0.722913i	$0.742803\pi$
$282$	0	0
$283$	22.8411	1.35776	0.678882	−	0.734248i	$-0.262465\pi$
$283$	22.8411	1.35776	0.678882	+	0.734248i	$0.262465\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.64556	−0.0971342
$288$	0	0
$289$	−7.31968	−0.430569
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.1041	−1.11608	−0.558038	−	0.829816i	$-0.688446\pi$
$293$	−19.1041	−1.11608	−0.558038	+	0.829816i	$0.688446\pi$
$294$	0	0
$295$	−40.2891	−2.34572
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.4318	−1.06594
$300$	0	0
$301$	−11.0402	−0.636348
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.4360	0.654823
$306$	0	0
$307$	19.0160	1.08530	0.542651	−	0.839958i	$-0.317421\pi$
$307$	19.0160	1.08530	0.542651	+	0.839958i	$0.317421\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.8722	0.843325	0.421662	−	0.906753i	$-0.361447\pi$
$311$	14.8722	0.843325	0.421662	+	0.906753i	$0.361447\pi$
$312$	0	0
$313$	23.3817	1.32161	0.660807	−	0.750556i	$-0.270214\pi$
$313$	23.3817	1.32161	0.660807	+	0.750556i	$0.270214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.1390	−1.18728	−0.593641	−	0.804730i	$-0.702310\pi$
$317$	−21.1390	−1.18728	−0.593641	+	0.804730i	$0.702310\pi$
$318$	0	0
$319$	1.79538	0.100522
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.98810	−0.166262
$324$	0	0
$325$	−51.6029	−2.86242
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.6896	−0.589336
$330$	0	0
$331$	−24.9220	−1.36984	−0.684918	−	0.728620i	$-0.740162\pi$
$331$	−24.9220	−1.36984	−0.684918	+	0.728620i	$0.740162\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−24.2759	−1.32634
$336$	0	0
$337$	−16.3352	−0.889833	−0.444916	−	0.895572i	$-0.646767\pi$
$337$	−16.3352	−0.889833	−0.444916	+	0.895572i	$0.646767\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	23.5147	1.27339
$342$	0	0
$343$	19.9339	1.07633
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.48755	−0.509318	−0.254659	−	0.967031i	$-0.581963\pi$
$347$	−9.48755	−0.509318	−0.254659	+	0.967031i	$0.581963\pi$
$348$	0	0
$349$	−29.7560	−1.59280	−0.796400	−	0.604770i	$-0.793265\pi$
$349$	−29.7560	−1.59280	−0.796400	+	0.604770i	$0.793265\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.5570	−0.774793	−0.387396	−	0.921913i	$-0.626625\pi$
$353$	−14.5570	−0.774793	−0.387396	+	0.921913i	$0.626625\pi$
$354$	0	0
$355$	−52.6937	−2.79669
$356$	0	0
$357$	0	0
$358$	0	0
$359$	24.0934	1.27160	0.635801	−	0.771853i	$-0.280670\pi$
$359$	24.0934	1.27160	0.635801	+	0.771853i	$0.280670\pi$
$360$	0	0
$361$	−18.0776	−0.951455
$362$	0	0
$363$	0	0
$364$	0	0
$365$	19.9983	1.04676
$366$	0	0
$367$	1.81012	0.0944873	0.0472437	−	0.998883i	$-0.484956\pi$
$367$	1.81012	0.0944873	0.0472437	+	0.998883i	$0.484956\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.2991	−0.534703
$372$	0	0
$373$	−18.5131	−0.958572	−0.479286	−	0.877659i	$-0.659104\pi$
$373$	−18.5131	−0.958572	−0.479286	+	0.877659i	$0.659104\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.08546	−0.158909
$378$	0	0
$379$	13.9743	0.717813	0.358906	−	0.933374i	$-0.383150\pi$
$379$	13.9743	0.717813	0.358906	+	0.933374i	$0.383150\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.77104	−0.345984	−0.172992	−	0.984923i	$-0.555343\pi$
$383$	−6.77104	−0.345984	−0.172992	+	0.984923i	$0.555343\pi$
$384$	0	0
$385$	−20.2010	−1.02954
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−11.0099	−0.558224	−0.279112	−	0.960259i	$-0.590040\pi$
$389$	−11.0099	−0.558224	−0.279112	+	0.960259i	$0.590040\pi$
$390$	0	0
$391$	13.4588	0.680639
$392$	0	0
$393$	0	0
$394$	0	0
$395$	24.1427	1.21475
$396$	0	0
$397$	−7.85407	−0.394185	−0.197092	−	0.980385i	$-0.563150\pi$
$397$	−7.85407	−0.394185	−0.197092	+	0.980385i	$0.563150\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	37.4096	1.86814	0.934072	−	0.357084i	$-0.116229\pi$
$401$	37.4096	1.86814	0.934072	+	0.357084i	$0.116229\pi$
$402$	0	0
$403$	−40.4113	−2.01303
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.6369	−1.27077
$408$	0	0
$409$	33.5533	1.65910	0.829551	−	0.558430i	$-0.188596\pi$
$409$	33.5533	1.65910	0.829551	+	0.558430i	$0.188596\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	19.1844	0.944005
$414$	0	0
$415$	56.9809	2.79708
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.23719	−0.451266	−0.225633	−	0.974212i	$-0.572445\pi$
$419$	−9.23719	−0.451266	−0.225633	+	0.974212i	$0.572445\pi$
$420$	0	0
$421$	−5.09928	−0.248524	−0.124262	−	0.992249i	$-0.539656\pi$
$421$	−5.09928	−0.248524	−0.124262	+	0.992249i	$0.539656\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	37.6802	1.82776
$426$	0	0
$427$	−5.44547	−0.263525
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.25936	−0.205166	−0.102583	−	0.994724i	$-0.532711\pi$
$431$	−4.25936	−0.205166	−0.102583	+	0.994724i	$0.532711\pi$
$432$	0	0
$433$	−23.5901	−1.13367	−0.566834	−	0.823832i	$-0.691832\pi$
$433$	−23.5901	−1.13367	−0.566834	+	0.823832i	$0.691832\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.15442	−0.198733
$438$	0	0
$439$	1.56456	0.0746722	0.0373361	−	0.999303i	$-0.488113\pi$
$439$	1.56456	0.0746722	0.0373361	+	0.999303i	$0.488113\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.8252	1.08446	0.542230	−	0.840230i	$-0.317580\pi$
$443$	22.8252	1.08446	0.542230	+	0.840230i	$0.317580\pi$
$444$	0	0
$445$	60.1919	2.85337
$446$	0	0
$447$	0	0
$448$	0	0
$449$	16.2906	0.768801	0.384401	−	0.923166i	$-0.374408\pi$
$449$	16.2906	0.768801	0.384401	+	0.923166i	$0.374408\pi$
$450$	0	0
$451$	2.07139	0.0975381
$452$	0	0
$453$	0	0
$454$	0	0
$455$	34.7164	1.62753
$456$	0	0
$457$	−1.71657	−0.0802976	−0.0401488	−	0.999194i	$-0.512783\pi$
$457$	−1.71657	−0.0802976	−0.0401488	+	0.999194i	$0.512783\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.0452	1.63222	0.816109	−	0.577898i	$-0.196127\pi$
$461$	35.0452	1.63222	0.816109	+	0.577898i	$0.196127\pi$
$462$	0	0
$463$	−9.87067	−0.458729	−0.229364	−	0.973341i	$-0.573665\pi$
$463$	−9.87067	−0.458729	−0.229364	+	0.973341i	$0.573665\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.86372	0.271341	0.135670	−	0.990754i	$-0.456681\pi$
$467$	5.86372	0.271341	0.135670	+	0.990754i	$0.456681\pi$
$468$	0	0
$469$	11.5595	0.533766
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.8972	0.638994
$474$	0	0
$475$	−11.6310	−0.533668
$476$	0	0
$477$	0	0
$478$	0	0
$479$	25.2665	1.15446	0.577228	−	0.816583i	$-0.304134\pi$
$479$	25.2665	1.15446	0.577228	+	0.816583i	$0.304134\pi$
$480$	0	0
$481$	44.0583	2.00888
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.0971	−0.458486
$486$	0	0
$487$	19.4718	0.882352	0.441176	−	0.897421i	$-0.354562\pi$
$487$	19.4718	0.882352	0.441176	+	0.897421i	$0.354562\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.5127	0.654948	0.327474	−	0.944860i	$-0.393803\pi$
$491$	14.5127	0.654948	0.327474	+	0.944860i	$0.393803\pi$
$492$	0	0
$493$	2.25299	0.101469
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.0911	1.12549
$498$	0	0
$499$	25.5687	1.14461	0.572305	−	0.820041i	$-0.306049\pi$
$499$	25.5687	1.14461	0.572305	+	0.820041i	$0.306049\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.01751	−0.179132	−0.0895659	−	0.995981i	$-0.528548\pi$
$503$	−4.01751	−0.179132	−0.0895659	+	0.995981i	$0.528548\pi$
$504$	0	0
$505$	−65.7189	−2.92445
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.0977	−1.42271	−0.711354	−	0.702834i	$-0.751918\pi$
$509$	−32.0977	−1.42271	−0.711354	+	0.702834i	$0.751918\pi$
$510$	0	0
$511$	−9.52259	−0.421254
$512$	0	0
$513$	0	0
$514$	0	0
$515$	21.3074	0.938914
$516$	0	0
$517$	13.4558	0.591786
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.3141	−1.19665	−0.598327	−	0.801252i	$-0.704168\pi$
$521$	−27.3141	−1.19665	−0.598327	+	0.801252i	$0.704168\pi$
$522$	0	0
$523$	18.5339	0.810433	0.405216	−	0.914221i	$-0.367196\pi$
$523$	18.5339	0.810433	0.405216	+	0.914221i	$0.367196\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.5081	1.28539
$528$	0	0
$529$	−4.28799	−0.186434
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.55980	−0.154192
$534$	0	0
$535$	62.4415	2.69958
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7.73662	−0.333240
$540$	0	0
$541$	10.2563	0.440954	0.220477	−	0.975392i	$-0.429239\pi$
$541$	10.2563	0.440954	0.220477	+	0.975392i	$0.429239\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−22.9147	−0.981558
$546$	0	0
$547$	−9.26173	−0.396003	−0.198001	−	0.980202i	$-0.563445\pi$
$547$	−9.26173	−0.396003	−0.198001	+	0.980202i	$0.563445\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.695446	−0.0296270
$552$	0	0
$553$	−11.4960	−0.488860
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.9335	1.81915	0.909576	−	0.415537i	$-0.136406\pi$
$557$	42.9335	1.81915	0.909576	+	0.415537i	$0.136406\pi$
$558$	0	0
$559$	−23.8831	−1.01015
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.8022	−0.623840	−0.311920	−	0.950108i	$-0.600972\pi$
$563$	−14.8022	−0.623840	−0.311920	+	0.950108i	$0.600972\pi$
$564$	0	0
$565$	−82.9639	−3.49032
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.2344	−1.93825	−0.969124	−	0.246574i	$-0.920695\pi$
$569$	−46.2344	−1.93825	−0.969124	+	0.246574i	$0.920695\pi$
$570$	0	0
$571$	−33.4665	−1.40053	−0.700264	−	0.713884i	$-0.746934\pi$
$571$	−33.4665	−1.40053	−0.700264	+	0.713884i	$0.746934\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	52.3876	2.18471
$576$	0	0
$577$	31.6747	1.31864	0.659319	−	0.751864i	$-0.270845\pi$
$577$	31.6747	1.31864	0.659319	+	0.751864i	$0.270845\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−27.1325	−1.12565
$582$	0	0
$583$	12.9643	0.536926
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.40965	−0.0581826	−0.0290913	−	0.999577i	$-0.509261\pi$
$587$	−1.40965	−0.0581826	−0.0290913	+	0.999577i	$0.509261\pi$
$588$	0	0
$589$	−9.10847	−0.375308
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−27.5495	−1.13132	−0.565661	−	0.824638i	$-0.691379\pi$
$593$	−27.5495	−1.13132	−0.565661	+	0.824638i	$0.691379\pi$
$594$	0	0
$595$	−25.3497	−1.03924
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.6171	0.842391	0.421196	−	0.906970i	$-0.361611\pi$
$599$	20.6171	0.842391	0.421196	+	0.906970i	$0.361611\pi$
$600$	0	0
$601$	−29.6549	−1.20965	−0.604825	−	0.796358i	$-0.706757\pi$
$601$	−29.6549	−1.20965	−0.604825	+	0.796358i	$0.706757\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20.0730	−0.816085
$606$	0	0
$607$	42.3384	1.71846	0.859232	−	0.511586i	$-0.170942\pi$
$607$	42.3384	1.71846	0.859232	+	0.511586i	$0.170942\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−23.1245	−0.935518
$612$	0	0
$613$	14.1001	0.569498	0.284749	−	0.958602i	$-0.408090\pi$
$613$	14.1001	0.569498	0.284749	+	0.958602i	$0.408090\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16.9805	−0.683611	−0.341806	−	0.939771i	$-0.611038\pi$
$617$	−16.9805	−0.683611	−0.341806	+	0.939771i	$0.611038\pi$
$618$	0	0
$619$	−22.5659	−0.907000	−0.453500	−	0.891256i	$-0.649825\pi$
$619$	−22.5659	−0.907000	−0.453500	+	0.891256i	$0.649825\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−28.6615	−1.14830
$624$	0	0
$625$	61.1149	2.44460
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−32.1711	−1.28275
$630$	0	0
$631$	−35.0492	−1.39529	−0.697644	−	0.716444i	$-0.745768\pi$
$631$	−35.0492	−1.39529	−0.697644	+	0.716444i	$0.745768\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21.3006	−0.845289
$636$	0	0
$637$	13.2958	0.526798
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−17.7215	−0.699955	−0.349978	−	0.936758i	$-0.613811\pi$
$641$	−17.7215	−0.699955	−0.349978	+	0.936758i	$0.613811\pi$
$642$	0	0
$643$	−8.78855	−0.346587	−0.173293	−	0.984870i	$-0.555441\pi$
$643$	−8.78855	−0.346587	−0.173293	+	0.984870i	$0.555441\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23.6010	0.927849	0.463925	−	0.885875i	$-0.346441\pi$
$647$	23.6010	0.927849	0.463925	+	0.885875i	$0.346441\pi$
$648$	0	0
$649$	−24.1490	−0.947930
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−25.3303	−0.991251	−0.495625	−	0.868536i	$-0.665061\pi$
$653$	−25.3303	−0.991251	−0.495625	+	0.868536i	$0.665061\pi$
$654$	0	0
$655$	36.0775	1.40967
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.0085	0.389876	0.194938	−	0.980816i	$-0.437550\pi$
$659$	10.0085	0.389876	0.194938	+	0.980816i	$0.437550\pi$
$660$	0	0
$661$	28.2211	1.09768	0.548838	−	0.835929i	$-0.315071\pi$
$661$	28.2211	1.09768	0.548838	+	0.835929i	$0.315071\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.82488	0.303436
$666$	0	0
$667$	3.13238	0.121286
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.85464	0.264620
$672$	0	0
$673$	−31.6135	−1.21861	−0.609306	−	0.792935i	$-0.708552\pi$
$673$	−31.6135	−1.21861	−0.609306	+	0.792935i	$0.708552\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.6518	0.409381	0.204690	−	0.978827i	$-0.434381\pi$
$677$	10.6518	0.409381	0.204690	+	0.978827i	$0.434381\pi$
$678$	0	0
$679$	4.80793	0.184511
$680$	0	0
$681$	0	0
$682$	0	0
$683$	43.3219	1.65766	0.828832	−	0.559497i	$-0.189006\pi$
$683$	43.3219	1.65766	0.828832	+	0.559497i	$0.189006\pi$
$684$	0	0
$685$	7.05485	0.269552
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−22.2798	−0.848794
$690$	0	0
$691$	−14.3875	−0.547326	−0.273663	−	0.961826i	$-0.588235\pi$
$691$	−14.3875	−0.547326	−0.273663	+	0.961826i	$0.588235\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	93.5281	3.54772
$696$	0	0
$697$	2.59934	0.0984571
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.54714	−0.322821	−0.161411	−	0.986887i	$-0.551604\pi$
$701$	−8.54714	−0.322821	−0.161411	+	0.986887i	$0.551604\pi$
$702$	0	0
$703$	9.93049	0.374536
$704$	0	0
$705$	0	0
$706$	0	0
$707$	31.2933	1.17691
$708$	0	0
$709$	−0.965486	−0.0362596	−0.0181298	−	0.999836i	$-0.505771\pi$
$709$	−0.965486	−0.0362596	−0.0181298	+	0.999836i	$0.505771\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	41.0257	1.53642
$714$	0	0
$715$	−43.7002	−1.63430
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.0744	0.413004	0.206502	−	0.978446i	$-0.433792\pi$
$719$	11.0744	0.413004	0.206502	+	0.978446i	$0.433792\pi$
$720$	0	0
$721$	−10.1459	−0.377853
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.76964	0.325696
$726$	0	0
$727$	−23.7985	−0.882638	−0.441319	−	0.897350i	$-0.645489\pi$
$727$	−23.7985	−0.882638	−0.441319	+	0.897350i	$0.645489\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.4393	0.645015
$732$	0	0
$733$	1.56039	0.0576342	0.0288171	−	0.999585i	$-0.490826\pi$
$733$	1.56039	0.0576342	0.0288171	+	0.999585i	$0.490826\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.5508	−0.535985
$738$	0	0
$739$	−38.6650	−1.42231	−0.711157	−	0.703033i	$-0.751828\pi$
$739$	−38.6650	−1.42231	−0.711157	+	0.703033i	$0.751828\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.6028	−1.26945	−0.634726	−	0.772737i	$-0.718887\pi$
$743$	−34.6028	−1.26945	−0.634726	+	0.772737i	$0.718887\pi$
$744$	0	0
$745$	47.4224	1.73742
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−29.7327	−1.08641
$750$	0	0
$751$	−24.4696	−0.892908	−0.446454	−	0.894807i	$-0.647313\pi$
$751$	−24.4696	−0.892908	−0.446454	+	0.894807i	$0.647313\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.2935	0.447408
$756$	0	0
$757$	39.9968	1.45371	0.726854	−	0.686793i	$-0.240982\pi$
$757$	39.9968	1.45371	0.726854	+	0.686793i	$0.240982\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.1342	−1.09236	−0.546182	−	0.837667i	$-0.683919\pi$
$761$	−30.1342	−1.09236	−0.546182	+	0.837667i	$0.683919\pi$
$762$	0	0
$763$	10.9113	0.395015
$764$	0	0
$765$	0	0
$766$	0	0
$767$	41.5012	1.49852
$768$	0	0
$769$	−3.23445	−0.116637	−0.0583186	−	0.998298i	$-0.518574\pi$
$769$	−3.23445	−0.116637	−0.0583186	+	0.998298i	$0.518574\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.10656	−0.0398002	−0.0199001	−	0.999802i	$-0.506335\pi$
$773$	−1.10656	−0.0398002	−0.0199001	+	0.999802i	$0.506335\pi$
$774$	0	0
$775$	114.859	4.12585
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.802358	−0.0287475
$780$	0	0
$781$	−31.5841	−1.13017
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.7781	0.955752
$786$	0	0
$787$	−53.1254	−1.89372	−0.946858	−	0.321650i	$-0.895762\pi$
$787$	−53.1254	−1.89372	−0.946858	+	0.321650i	$0.895762\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	39.5049	1.40463
$792$	0	0
$793$	−11.7800	−0.418322
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21.5294	0.762611	0.381306	−	0.924449i	$-0.375475\pi$
$797$	21.5294	0.762611	0.381306	+	0.924449i	$0.375475\pi$
$798$	0	0
$799$	16.8854	0.597362
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.9868	0.423006
$804$	0	0
$805$	−35.2443	−1.24220
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.7019	1.50132	0.750660	−	0.660688i	$-0.229736\pi$
$809$	42.7019	1.50132	0.750660	+	0.660688i	$0.229736\pi$
$810$	0	0
$811$	6.75273	0.237121	0.118560	−	0.992947i	$-0.462172\pi$
$811$	6.75273	0.237121	0.118560	+	0.992947i	$0.462172\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−75.8592	−2.65723
$816$	0	0
$817$	−5.38311	−0.188331
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.0124	−0.733337	−0.366668	−	0.930352i	$-0.619502\pi$
$821$	−21.0124	−0.733337	−0.366668	+	0.930352i	$0.619502\pi$
$822$	0	0
$823$	37.9581	1.32314	0.661569	−	0.749884i	$-0.269891\pi$
$823$	37.9581	1.32314	0.661569	+	0.749884i	$0.269891\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.0210	1.07871	0.539353	−	0.842080i	$-0.318669\pi$
$827$	31.0210	1.07871	0.539353	+	0.842080i	$0.318669\pi$
$828$	0	0
$829$	−21.3531	−0.741622	−0.370811	−	0.928708i	$-0.620920\pi$
$829$	−21.3531	−0.741622	−0.370811	+	0.928708i	$0.620920\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9.70850	−0.336380
$834$	0	0
$835$	57.4609	1.98852
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.4426	−1.43076	−0.715379	−	0.698737i	$-0.753746\pi$
$839$	−41.4426	−1.43076	−0.715379	+	0.698737i	$0.753746\pi$
$840$	0	0
$841$	−28.4756	−0.981919
$842$	0	0
$843$	0	0
$844$	0	0
$845$	21.3265	0.733655
$846$	0	0
$847$	9.55816	0.328422
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−44.7282	−1.53326
$852$	0	0
$853$	−38.8606	−1.33056	−0.665281	−	0.746593i	$-0.731688\pi$
$853$	−38.8606	−1.33056	−0.665281	+	0.746593i	$0.731688\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.11511	0.311366	0.155683	−	0.987807i	$-0.450242\pi$
$857$	9.11511	0.311366	0.155683	+	0.987807i	$0.450242\pi$
$858$	0	0
$859$	1.71506	0.0585171	0.0292585	−	0.999572i	$-0.490685\pi$
$859$	1.71506	0.0585171	0.0292585	+	0.999572i	$0.490685\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10.5773	−0.360054	−0.180027	−	0.983662i	$-0.557618\pi$
$863$	−10.5773	−0.360054	−0.180027	+	0.983662i	$0.557618\pi$
$864$	0	0
$865$	103.074	3.50461
$866$	0	0
$867$	0	0
$868$	0	0
$869$	14.4709	0.490892
$870$	0	0
$871$	25.0063	0.847305
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−57.9347	−1.95855
$876$	0	0
$877$	54.3570	1.83551	0.917753	−	0.397151i	$-0.130001\pi$
$877$	54.3570	1.83551	0.917753	+	0.397151i	$0.130001\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17.5910	−0.592655	−0.296328	−	0.955086i	$-0.595762\pi$
$881$	−17.5910	−0.592655	−0.296328	+	0.955086i	$0.595762\pi$
$882$	0	0
$883$	16.5786	0.557914	0.278957	−	0.960304i	$-0.410011\pi$
$883$	16.5786	0.557914	0.278957	+	0.960304i	$0.410011\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.0105	−0.739040	−0.369520	−	0.929223i	$-0.620478\pi$
$887$	−22.0105	−0.739040	−0.369520	+	0.929223i	$0.620478\pi$
$888$	0	0
$889$	10.1427	0.340175
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.21214	−0.174418
$894$	0	0
$895$	12.3101	0.411480
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.86767	0.229050
$900$	0	0
$901$	16.2686	0.541986
$902$	0	0
$903$	0	0
$904$	0	0
$905$	85.4297	2.83978
$906$	0	0
$907$	−8.98058	−0.298195	−0.149098	−	0.988822i	$-0.547637\pi$
$907$	−8.98058	−0.298195	−0.149098	+	0.988822i	$0.547637\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−56.4895	−1.87158	−0.935791	−	0.352556i	$-0.885313\pi$
$911$	−56.4895	−1.87158	−0.935791	+	0.352556i	$0.885313\pi$
$912$	0	0
$913$	34.1538	1.13033
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.1790	−0.567301
$918$	0	0
$919$	−26.5270	−0.875046	−0.437523	−	0.899207i	$-0.644144\pi$
$919$	−26.5270	−0.875046	−0.437523	+	0.899207i	$0.644144\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	54.2790	1.78661
$924$	0	0
$925$	−125.224	−4.11735
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−12.9236	−0.424011	−0.212006	−	0.977268i	$-0.568000\pi$
$929$	−12.9236	−0.424011	−0.212006	+	0.977268i	$0.568000\pi$
$930$	0	0
$931$	2.99679	0.0982160
$932$	0	0
$933$	0	0
$934$	0	0
$935$	31.9097	1.04356
$936$	0	0
$937$	−15.2432	−0.497974	−0.248987	−	0.968507i	$-0.580098\pi$
$937$	−15.2432	−0.497974	−0.248987	+	0.968507i	$0.580098\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52.9385	1.72575	0.862873	−	0.505421i	$-0.168663\pi$
$941$	52.9385	1.72575	0.862873	+	0.505421i	$0.168663\pi$
$942$	0	0
$943$	3.61392	0.117686
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.5444	0.992560	0.496280	−	0.868162i	$-0.334699\pi$
$947$	30.5444	0.992560	0.496280	+	0.868162i	$0.334699\pi$
$948$	0	0
$949$	−20.6000	−0.668703
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21.3202	−0.690631	−0.345315	−	0.938487i	$-0.612228\pi$
$953$	−21.3202	−0.690631	−0.345315	+	0.938487i	$0.612228\pi$
$954$	0	0
$955$	−10.9709	−0.355009
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.35930	−0.108477
$960$	0	0
$961$	58.9481	1.90155
$962$	0	0
$963$	0	0
$964$	0	0
$965$	35.7514	1.15088
$966$	0	0
$967$	20.5426	0.660607	0.330303	−	0.943875i	$-0.392849\pi$
$967$	20.5426	0.660607	0.330303	+	0.943875i	$0.392849\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.4845	0.529013	0.264507	−	0.964384i	$-0.414791\pi$
$971$	16.4845	0.529013	0.264507	+	0.964384i	$0.414791\pi$
$972$	0	0
$973$	−44.5352	−1.42773
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.79369	0.217350	0.108675	−	0.994077i	$-0.465339\pi$
$977$	6.79369	0.217350	0.108675	+	0.994077i	$0.465339\pi$
$978$	0	0
$979$	36.0785	1.15307
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.7642	−0.821752	−0.410876	−	0.911691i	$-0.634777\pi$
$983$	−25.7642	−0.821752	−0.410876	+	0.911691i	$0.634777\pi$
$984$	0	0
$985$	−2.94673	−0.0938907
$986$	0	0
$987$	0	0
$988$	0	0
$989$	24.2462	0.770984
$990$	0	0
$991$	18.9580	0.602220	0.301110	−	0.953589i	$-0.402643\pi$
$991$	18.9580	0.602220	0.301110	+	0.953589i	$0.402643\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	43.2275	1.37040
$996$	0	0
$997$	11.7748	0.372912	0.186456	−	0.982463i	$-0.440300\pi$
$997$	11.7748	0.372912	0.186456	+	0.982463i	$0.440300\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.j.1.7		7
3.2	odd	2		892.2.a.d.1.7	✓	7
12.11	even	2		3568.2.a.m.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
892.2.a.d.1.7	✓	7	3.2	odd	2
3568.2.a.m.1.1		7	12.11	even	2
8028.2.a.j.1.7		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+4.13650 q^{5} -1.96968 q^{7} +O(q^{10})\)	\(q+4.13650 q^{5} -1.96968 q^{7} +2.47938 q^{11} -4.26095 q^{13} +3.11132 q^{17} -0.960395 q^{19} +4.32574 q^{23} +12.1107 q^{25} +0.724125 q^{29} +9.48409 q^{31} -8.14757 q^{35} -10.3400 q^{37} +0.835446 q^{41} +5.60510 q^{43} +5.42708 q^{47} -3.12038 q^{49} +5.22884 q^{53} +10.2560 q^{55} -9.73990 q^{59} +2.76465 q^{61} -17.6254 q^{65} -5.86871 q^{67} -12.7387 q^{71} +4.83460 q^{73} -4.88358 q^{77} +5.83650 q^{79} +13.7751 q^{83} +12.8700 q^{85} +14.5514 q^{89} +8.39269 q^{91} -3.97268 q^{95} -2.44098 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{5} - q^{7}+O(q^{10}) \) 7 * q + 7 * q^5 - q^7	\( 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) \) 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

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