LMFDB - Embedded newform 4788.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.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:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.321637$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.21819 q^{5} +1.00000 q^{7} -2.00000 q^{11} +0.643274 q^{13} +4.21819 q^{17} -1.00000 q^{19} -2.00000 q^{23} +12.7931 q^{25} -4.93164 q^{29} +1.35673 q^{31} +4.21819 q^{35} +10.4364 q^{37} +4.00000 q^{41} -5.79310 q^{43} -5.50474 q^{47} +1.00000 q^{49} -6.21819 q^{53} -8.43637 q^{55} +14.4364 q^{61} +2.71345 q^{65} +8.43637 q^{67} +6.21819 q^{71} +3.28655 q^{73} -2.00000 q^{77} +15.1498 q^{79} -2.93164 q^{83} +17.7931 q^{85} +4.00000 q^{89} +0.643274 q^{91} -4.21819 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{7} - 6 q^{11} + 2 q^{13} - 3 q^{19} - 6 q^{23} + 13 q^{25} - 2 q^{29} + 4 q^{31} + 6 q^{37} + 12 q^{41} + 8 q^{43} - 4 q^{47} + 3 q^{49} - 6 q^{53} + 18 q^{61} + 8 q^{65} + 6 q^{71} + 10 q^{73}+ \cdots - 6 q^{97}+O(q^{100}) $ 3 * q + 3 * q^7 - 6 * q^11 + 2 * q^13 - 3 * q^19 - 6 * q^23 + 13 * q^25 - 2 * q^29 + 4 * q^31 + 6 * q^37 + 12 * q^41 + 8 * q^43 - 4 * q^47 + 3 * q^49 - 6 * q^53 + 18 * q^61 + 8 * q^65 + 6 * q^71 + 10 * q^73 - 6 * q^77 + 20 * q^79 + 4 * q^83 + 28 * q^85 + 12 * q^89 + 2 * q^91 - 6 * q^97

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
$5$	4.21819	1.88643	0.943215	−	0.332182i	$-0.107785\pi$
$5$	4.21819	1.88643	0.943215	+	0.332182i	$0.107785\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.643274	0.178412	0.0892061	−	0.996013i	$-0.471567\pi$
$13$	0.643274	0.178412	0.0892061	+	0.996013i	$0.471567\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.21819	1.02306	0.511530	−	0.859265i	$-0.329079\pi$
$17$	4.21819	1.02306	0.511530	+	0.859265i	$0.329079\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	12.7931	2.55862
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.93164	−0.915782	−0.457891	−	0.889008i	$-0.651395\pi$
$29$	−4.93164	−0.915782	−0.457891	+	0.889008i	$0.651395\pi$
$30$	0	0
$31$	1.35673	0.243675	0.121838	−	0.992550i	$-0.461121\pi$
$31$	1.35673	0.243675	0.121838	+	0.992550i	$0.461121\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.21819	0.713004
$36$	0	0
$37$	10.4364	1.71573	0.857865	−	0.513876i	$-0.171791\pi$
$37$	10.4364	1.71573	0.857865	+	0.513876i	$0.171791\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−5.79310	−0.883439	−0.441720	−	0.897153i	$-0.645631\pi$
$43$	−5.79310	−0.883439	−0.441720	+	0.897153i	$0.645631\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.50474	−0.802948	−0.401474	−	0.915870i	$-0.631502\pi$
$47$	−5.50474	−0.802948	−0.401474	+	0.915870i	$0.631502\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.21819	−0.854134	−0.427067	−	0.904220i	$-0.640453\pi$
$53$	−6.21819	−0.854134	−0.427067	+	0.904220i	$0.640453\pi$
$54$	0	0
$55$	−8.43637	−1.13756
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	14.4364	1.84839	0.924194	−	0.381923i	$-0.124738\pi$
$61$	14.4364	1.84839	0.924194	+	0.381923i	$0.124738\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.71345	0.336562
$66$	0	0
$67$	8.43637	1.03067	0.515334	−	0.856990i	$-0.327668\pi$
$67$	8.43637	1.03067	0.515334	+	0.856990i	$0.327668\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.21819	0.737963	0.368981	−	0.929437i	$-0.379707\pi$
$71$	6.21819	0.737963	0.368981	+	0.929437i	$0.379707\pi$
$72$	0	0
$73$	3.28655	0.384661	0.192331	−	0.981330i	$-0.438395\pi$
$73$	3.28655	0.384661	0.192331	+	0.981330i	$0.438395\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	15.1498	1.70449	0.852244	−	0.523144i	$-0.175241\pi$
$79$	15.1498	1.70449	0.852244	+	0.523144i	$0.175241\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.93164	−0.321789	−0.160895	−	0.986972i	$-0.551438\pi$
$83$	−2.93164	−0.321789	−0.160895	+	0.986972i	$0.551438\pi$
$84$	0	0
$85$	17.7931	1.92993
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0.643274	0.0674335
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.21819	−0.432777
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.21819	−0.817740	−0.408870	−	0.912593i	$-0.634077\pi$
$101$	−8.21819	−0.817740	−0.408870	+	0.912593i	$0.634077\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.3680	1.29233	0.646167	−	0.763196i	$-0.276371\pi$
$107$	13.3680	1.29233	0.646167	+	0.763196i	$0.276371\pi$
$108$	0	0
$109$	6.43637	0.616493	0.308246	−	0.951307i	$-0.400258\pi$
$109$	6.43637	0.616493	0.308246	+	0.951307i	$0.400258\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.50474	−0.329698	−0.164849	−	0.986319i	$-0.552714\pi$
$113$	−3.50474	−0.329698	−0.164849	+	0.986319i	$0.552714\pi$
$114$	0	0
$115$	−8.43637	−0.786696
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.21819	0.386681
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	32.8727	2.94023
$126$	0	0
$127$	−4.43637	−0.393664	−0.196832	−	0.980437i	$-0.563065\pi$
$127$	−4.43637	−0.393664	−0.196832	+	0.980437i	$0.563065\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.3680	−1.69219	−0.846096	−	0.533031i	$-0.821053\pi$
$131$	−19.3680	−1.69219	−0.846096	+	0.533031i	$0.821053\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.28655	0.622532	0.311266	−	0.950323i	$-0.399247\pi$
$137$	7.28655	0.622532	0.311266	+	0.950323i	$0.399247\pi$
$138$	0	0
$139$	−18.1593	−1.54025	−0.770126	−	0.637892i	$-0.779807\pi$
$139$	−18.1593	−1.54025	−0.770126	+	0.637892i	$0.779807\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.28655	−0.107587
$144$	0	0
$145$	−20.8026	−1.72756
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.14982	−0.421890	−0.210945	−	0.977498i	$-0.567654\pi$
$149$	−5.14982	−0.421890	−0.210945	+	0.977498i	$0.567654\pi$
$150$	0	0
$151$	−2.71345	−0.220818	−0.110409	−	0.993886i	$-0.535216\pi$
$151$	−2.71345	−0.220818	−0.110409	+	0.993886i	$0.535216\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.72292	0.459676
$156$	0	0
$157$	−7.72292	−0.616356	−0.308178	−	0.951329i	$-0.599719\pi$
$157$	−7.72292	−0.616356	−0.308178	+	0.951329i	$0.599719\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−4.50655	−0.352980	−0.176490	−	0.984302i	$-0.556474\pi$
$163$	−4.50655	−0.352980	−0.176490	+	0.984302i	$0.556474\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.1498	1.79139	0.895694	−	0.444672i	$-0.146680\pi$
$167$	23.1498	1.79139	0.895694	+	0.444672i	$0.146680\pi$
$168$	0	0
$169$	−12.5862	−0.968169
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.8727	−1.58693	−0.793463	−	0.608619i	$-0.791724\pi$
$173$	−20.8727	−1.58693	−0.793463	+	0.608619i	$0.791724\pi$
$174$	0	0
$175$	12.7931	0.967067
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−13.3680	−0.999172	−0.499586	−	0.866264i	$-0.666515\pi$
$179$	−13.3680	−0.999172	−0.499586	+	0.866264i	$0.666515\pi$
$180$	0	0
$181$	−8.71345	−0.647666	−0.323833	−	0.946114i	$-0.604972\pi$
$181$	−8.71345	−0.647666	−0.323833	+	0.946114i	$0.604972\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	44.0226	3.23660
$186$	0	0
$187$	−8.43637	−0.616929
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.713451	−0.0516235	−0.0258118	−	0.999667i	$-0.508217\pi$
$191$	−0.713451	−0.0516235	−0.0258118	+	0.999667i	$0.508217\pi$
$192$	0	0
$193$	11.7229	0.843834	0.421917	−	0.906634i	$-0.361357\pi$
$193$	11.7229	0.843834	0.421917	+	0.906634i	$0.361357\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	24.4364	1.73225	0.866124	−	0.499829i	$-0.166604\pi$
$199$	24.4364	1.73225	0.866124	+	0.499829i	$0.166604\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.93164	−0.346133
$204$	0	0
$205$	16.8727	1.17844
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−24.4364	−1.66655
$216$	0	0
$217$	1.35673	0.0921005
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.71345	0.182526
$222$	0	0
$223$	−28.8026	−1.92876	−0.964381	−	0.264516i	$-0.914788\pi$
$223$	−28.8026	−1.92876	−0.964381	+	0.264516i	$0.914788\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.27708	0.151135	0.0755675	−	0.997141i	$-0.475923\pi$
$227$	2.27708	0.151135	0.0755675	+	0.997141i	$0.475923\pi$
$228$	0	0
$229$	−26.0226	−1.71962	−0.859810	−	0.510614i	$-0.829418\pi$
$229$	−26.0226	−1.71962	−0.859810	+	0.510614i	$0.829418\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.0095	0.852279	0.426139	−	0.904658i	$-0.359873\pi$
$233$	13.0095	0.852279	0.426139	+	0.904658i	$0.359873\pi$
$234$	0	0
$235$	−23.2200	−1.51471
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.277078	0.0179227	0.00896134	−	0.999960i	$-0.497147\pi$
$239$	0.277078	0.0179227	0.00896134	+	0.999960i	$0.497147\pi$
$240$	0	0
$241$	17.5862	1.13283	0.566413	−	0.824121i	$-0.308331\pi$
$241$	17.5862	1.13283	0.566413	+	0.824121i	$0.308331\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.21819	0.269490
$246$	0	0
$247$	−0.643274	−0.0409306
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.49526	0.409977	0.204989	−	0.978764i	$-0.434284\pi$
$251$	6.49526	0.409977	0.204989	+	0.978764i	$0.434284\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.850175	0.0530325	0.0265162	−	0.999648i	$-0.491559\pi$
$257$	0.850175	0.0530325	0.0265162	+	0.999648i	$0.491559\pi$
$258$	0	0
$259$	10.4364	0.648485
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.4364	−0.643534	−0.321767	−	0.946819i	$-0.604277\pi$
$263$	−10.4364	−0.643534	−0.321767	+	0.946819i	$0.604277\pi$
$264$	0	0
$265$	−26.2295	−1.61126
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.4364	1.00214	0.501072	−	0.865406i	$-0.332939\pi$
$269$	16.4364	1.00214	0.501072	+	0.865406i	$0.332939\pi$
$270$	0	0
$271$	−12.0226	−0.730319	−0.365160	−	0.930945i	$-0.618986\pi$
$271$	−12.0226	−0.730319	−0.365160	+	0.930945i	$0.618986\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−25.5862	−1.54291
$276$	0	0
$277$	−15.6527	−0.940482	−0.470241	−	0.882538i	$-0.655833\pi$
$277$	−15.6527	−0.940482	−0.470241	+	0.882538i	$0.655833\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.2182	0.848186	0.424093	−	0.905619i	$-0.360593\pi$
$281$	14.2182	0.848186	0.424093	+	0.905619i	$0.360593\pi$
$282$	0	0
$283$	15.1498	0.900564	0.450282	−	0.892886i	$-0.351324\pi$
$283$	15.1498	0.900564	0.450282	+	0.892886i	$0.351324\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	0.793099	0.0466529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.4589	−1.42891	−0.714453	−	0.699683i	$-0.753325\pi$
$293$	−24.4589	−1.42891	−0.714453	+	0.699683i	$0.753325\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.28655	−0.0744030
$300$	0	0
$301$	−5.79310	−0.333909
$302$	0	0
$303$	0	0
$304$	0	0
$305$	60.8953	3.48686
$306$	0	0
$307$	28.3888	1.62023	0.810116	−	0.586269i	$-0.199404\pi$
$307$	28.3888	1.62023	0.810116	+	0.586269i	$0.199404\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−28.2408	−1.60139	−0.800693	−	0.599075i	$-0.795535\pi$
$311$	−28.2408	−1.60139	−0.800693	+	0.599075i	$0.795535\pi$
$312$	0	0
$313$	−34.0226	−1.92307	−0.961535	−	0.274683i	$-0.911427\pi$
$313$	−34.0226	−1.92307	−0.961535	+	0.274683i	$0.911427\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.07783	−0.341365	−0.170683	−	0.985326i	$-0.554597\pi$
$317$	−6.07783	−0.341365	−0.170683	+	0.985326i	$0.554597\pi$
$318$	0	0
$319$	9.86328	0.552237
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.21819	−0.234706
$324$	0	0
$325$	8.22947	0.456489
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.50474	−0.303486
$330$	0	0
$331$	28.8727	1.58699	0.793495	−	0.608577i	$-0.208259\pi$
$331$	28.8727	1.58699	0.793495	+	0.608577i	$0.208259\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	35.5862	1.94428
$336$	0	0
$337$	1.56363	0.0851762	0.0425881	−	0.999093i	$-0.486440\pi$
$337$	1.56363	0.0851762	0.0425881	+	0.999093i	$0.486440\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.71345	−0.146942
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.1498	−0.920651	−0.460325	−	0.887750i	$-0.652267\pi$
$347$	−17.1498	−0.920651	−0.460325	+	0.887750i	$0.652267\pi$
$348$	0	0
$349$	26.4589	1.41631	0.708157	−	0.706055i	$-0.249527\pi$
$349$	26.4589	1.41631	0.708157	+	0.706055i	$0.249527\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.7818	0.627083	0.313541	−	0.949575i	$-0.398485\pi$
$353$	11.7818	0.627083	0.313541	+	0.949575i	$0.398485\pi$
$354$	0	0
$355$	26.2295	1.39212
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.4589	1.18534	0.592669	−	0.805446i	$-0.298074\pi$
$359$	22.4589	1.18534	0.592669	+	0.805446i	$0.298074\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	13.8633	0.725637
$366$	0	0
$367$	−16.5767	−0.865298	−0.432649	−	0.901562i	$-0.642421\pi$
$367$	−16.5767	−0.865298	−0.432649	+	0.901562i	$0.642421\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.21819	−0.322832
$372$	0	0
$373$	−16.2996	−0.843964	−0.421982	−	0.906604i	$-0.638665\pi$
$373$	−16.2996	−0.843964	−0.421982	+	0.906604i	$0.638665\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.17240	−0.163387
$378$	0	0
$379$	21.0131	1.07937	0.539685	−	0.841867i	$-0.318543\pi$
$379$	21.0131	1.07937	0.539685	+	0.841867i	$0.318543\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.4364	−1.65742	−0.828711	−	0.559677i	$-0.810925\pi$
$383$	−32.4364	−1.65742	−0.828711	+	0.559677i	$0.810925\pi$
$384$	0	0
$385$	−8.43637	−0.429957
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.7360	1.65978	0.829891	−	0.557925i	$-0.188402\pi$
$389$	32.7360	1.65978	0.829891	+	0.557925i	$0.188402\pi$
$390$	0	0
$391$	−8.43637	−0.426646
$392$	0	0
$393$	0	0
$394$	0	0
$395$	63.9048	3.21540
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.7818	0.688231	0.344115	−	0.938927i	$-0.388179\pi$
$401$	13.7818	0.688231	0.344115	+	0.938927i	$0.388179\pi$
$402$	0	0
$403$	0.872747	0.0434746
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.8727	−1.03462
$408$	0	0
$409$	13.5160	0.668324	0.334162	−	0.942516i	$-0.391547\pi$
$409$	13.5160	0.668324	0.334162	+	0.942516i	$0.391547\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.3662	−0.607033
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−16.2408	−0.793413	−0.396706	−	0.917946i	$-0.629847\pi$
$419$	−16.2408	−0.793413	−0.396706	+	0.917946i	$0.629847\pi$
$420$	0	0
$421$	7.28655	0.355125	0.177562	−	0.984110i	$-0.443179\pi$
$421$	7.28655	0.355125	0.177562	+	0.984110i	$0.443179\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	53.9637	2.61762
$426$	0	0
$427$	14.4364	0.698625
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.8139	1.38791	0.693957	−	0.720016i	$-0.255866\pi$
$431$	28.8139	1.38791	0.693957	+	0.720016i	$0.255866\pi$
$432$	0	0
$433$	22.4589	1.07931	0.539654	−	0.841887i	$-0.318555\pi$
$433$	22.4589	1.07931	0.539654	+	0.841887i	$0.318555\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−23.6564	−1.12906	−0.564529	−	0.825414i	$-0.690942\pi$
$439$	−23.6564	−1.12906	−0.564529	+	0.825414i	$0.690942\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.5636	−0.644427	−0.322214	−	0.946667i	$-0.604427\pi$
$443$	−13.5636	−0.644427	−0.322214	+	0.946667i	$0.604427\pi$
$444$	0	0
$445$	16.8727	0.799845
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.5178	−0.968297	−0.484148	−	0.874986i	$-0.660870\pi$
$449$	−20.5178	−0.968297	−0.484148	+	0.874986i	$0.660870\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	0	0
$454$	0	0
$455$	2.71345	0.127209
$456$	0	0
$457$	−7.79310	−0.364546	−0.182273	−	0.983248i	$-0.558345\pi$
$457$	−7.79310	−0.364546	−0.182273	+	0.983248i	$0.558345\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.20872	0.242594	0.121297	−	0.992616i	$-0.461295\pi$
$461$	5.20872	0.242594	0.121297	+	0.992616i	$0.461295\pi$
$462$	0	0
$463$	9.42690	0.438105	0.219053	−	0.975713i	$-0.429703\pi$
$463$	9.42690	0.438105	0.219053	+	0.975713i	$0.429703\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.9411	−0.830216	−0.415108	−	0.909772i	$-0.636256\pi$
$467$	−17.9411	−0.830216	−0.415108	+	0.909772i	$0.636256\pi$
$468$	0	0
$469$	8.43637	0.389556
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.5862	0.532734
$474$	0	0
$475$	−12.7931	−0.586988
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.6677	−1.53831	−0.769157	−	0.639059i	$-0.779324\pi$
$479$	−33.6677	−1.53831	−0.769157	+	0.639059i	$0.779324\pi$
$480$	0	0
$481$	6.71345	0.306107
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.43637	−0.383076
$486$	0	0
$487$	−43.5862	−1.97508	−0.987540	−	0.157371i	$-0.949698\pi$
$487$	−43.5862	−1.97508	−0.987540	+	0.157371i	$0.949698\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	22.0226	0.993865	0.496932	−	0.867789i	$-0.334460\pi$
$491$	22.0226	0.993865	0.496932	+	0.867789i	$0.334460\pi$
$492$	0	0
$493$	−20.8026	−0.936901
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.21819	0.278924
$498$	0	0
$499$	−27.1724	−1.21640	−0.608202	−	0.793782i	$-0.708109\pi$
$499$	−27.1724	−1.21640	−0.608202	+	0.793782i	$0.708109\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	18.3775	0.819411	0.409706	−	0.912218i	$-0.365631\pi$
$503$	18.3775	0.819411	0.409706	+	0.912218i	$0.365631\pi$
$504$	0	0
$505$	−34.6658	−1.54261
$506$	0	0
$507$	0	0
$508$	0	0
$509$	11.0095	0.487986	0.243993	−	0.969777i	$-0.421543\pi$
$509$	11.0095	0.487986	0.243993	+	0.969777i	$0.421543\pi$
$510$	0	0
$511$	3.28655	0.145388
$512$	0	0
$513$	0	0
$514$	0	0
$515$	33.7455	1.48700
$516$	0	0
$517$	11.0095	0.484196
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.0095	1.53379	0.766896	−	0.641771i	$-0.221800\pi$
$521$	35.0095	1.53379	0.766896	+	0.641771i	$0.221800\pi$
$522$	0	0
$523$	−2.50292	−0.109445	−0.0547225	−	0.998502i	$-0.517427\pi$
$523$	−2.50292	−0.109445	−0.0547225	+	0.998502i	$0.517427\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.72292	0.249294
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.57310	0.111453
$534$	0	0
$535$	56.3888	2.43790
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	43.0131	1.84928	0.924639	−	0.380845i	$-0.124367\pi$
$541$	43.0131	1.84928	0.924639	+	0.380845i	$0.124367\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	27.1498	1.16297
$546$	0	0
$547$	−10.2771	−0.439416	−0.219708	−	0.975566i	$-0.570510\pi$
$547$	−10.2771	−0.439416	−0.219708	+	0.975566i	$0.570510\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.93164	0.210095
$552$	0	0
$553$	15.1498	0.644236
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.8502	−0.629222	−0.314611	−	0.949221i	$-0.601874\pi$
$557$	−14.8502	−0.629222	−0.314611	+	0.949221i	$0.601874\pi$
$558$	0	0
$559$	−3.72655	−0.157616
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−11.1498	−0.469909	−0.234955	−	0.972006i	$-0.575494\pi$
$563$	−11.1498	−0.469909	−0.234955	+	0.972006i	$0.575494\pi$
$564$	0	0
$565$	−14.7836	−0.621952
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−27.5273	−1.15400	−0.577002	−	0.816742i	$-0.695778\pi$
$569$	−27.5273	−1.15400	−0.577002	+	0.816742i	$0.695778\pi$
$570$	0	0
$571$	14.2996	0.598422	0.299211	−	0.954187i	$-0.403277\pi$
$571$	14.2996	0.598422	0.299211	+	0.954187i	$0.403277\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−25.5862	−1.06702
$576$	0	0
$577$	−16.2771	−0.677624	−0.338812	−	0.940854i	$-0.610025\pi$
$577$	−16.2771	−0.677624	−0.338812	+	0.940854i	$0.610025\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.93164	−0.121625
$582$	0	0
$583$	12.4364	0.515062
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.2408	1.33072	0.665359	−	0.746523i	$-0.268278\pi$
$587$	32.2408	1.33072	0.665359	+	0.746523i	$0.268278\pi$
$588$	0	0
$589$	−1.35673	−0.0559029
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.9447	1.31181	0.655907	−	0.754842i	$-0.272287\pi$
$593$	31.9447	1.31181	0.655907	+	0.754842i	$0.272287\pi$
$594$	0	0
$595$	17.7931	0.729446
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−32.8139	−1.34074	−0.670369	−	0.742028i	$-0.733864\pi$
$599$	−32.8139	−1.34074	−0.670369	+	0.742028i	$0.733864\pi$
$600$	0	0
$601$	−33.4458	−1.36428	−0.682142	−	0.731220i	$-0.738952\pi$
$601$	−33.4458	−1.36428	−0.682142	+	0.731220i	$0.738952\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−29.5273	−1.20046
$606$	0	0
$607$	5.42690	0.220271	0.110136	−	0.993917i	$-0.464871\pi$
$607$	5.42690	0.220271	0.110136	+	0.993917i	$0.464871\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.54106	−0.143256
$612$	0	0
$613$	6.92035	0.279510	0.139755	−	0.990186i	$-0.455368\pi$
$613$	6.92035	0.279510	0.139755	+	0.990186i	$0.455368\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−47.7455	−1.92216	−0.961081	−	0.276268i	$-0.910902\pi$
$617$	−47.7455	−1.92216	−0.961081	+	0.276268i	$0.910902\pi$
$618$	0	0
$619$	18.1593	0.729884	0.364942	−	0.931030i	$-0.381089\pi$
$619$	18.1593	0.729884	0.364942	+	0.931030i	$0.381089\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	74.6979	2.98792
$626$	0	0
$627$	0	0
$628$	0	0
$629$	44.0226	1.75529
$630$	0	0
$631$	19.9524	0.794292	0.397146	−	0.917755i	$-0.370001\pi$
$631$	19.9524	0.794292	0.397146	+	0.917755i	$0.370001\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18.7135	−0.742621
$636$	0	0
$637$	0.643274	0.0254875
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.7949	−1.05834	−0.529168	−	0.848517i	$-0.677496\pi$
$641$	−26.7949	−1.05834	−0.529168	+	0.848517i	$0.677496\pi$
$642$	0	0
$643$	−20.5767	−0.811467	−0.405733	−	0.913991i	$-0.632984\pi$
$643$	−20.5767	−0.811467	−0.405733	+	0.913991i	$0.632984\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.4953	0.412611	0.206306	−	0.978488i	$-0.433856\pi$
$647$	10.4953	0.412611	0.206306	+	0.978488i	$0.433856\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.87275	0.112419	0.0562096	−	0.998419i	$-0.482098\pi$
$653$	2.87275	0.112419	0.0562096	+	0.998419i	$0.482098\pi$
$654$	0	0
$655$	−81.6979	−3.19220
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.9542	0.660442	0.330221	−	0.943904i	$-0.392877\pi$
$659$	16.9542	0.660442	0.330221	+	0.943904i	$0.392877\pi$
$660$	0	0
$661$	13.3757	0.520253	0.260127	−	0.965575i	$-0.416236\pi$
$661$	13.3757	0.520253	0.260127	+	0.965575i	$0.416236\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.21819	−0.163574
$666$	0	0
$667$	9.86328	0.381908
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−28.8727	−1.11462
$672$	0	0
$673$	24.2771	0.935813	0.467906	−	0.883778i	$-0.345008\pi$
$673$	24.2771	0.935813	0.467906	+	0.883778i	$0.345008\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−43.6088	−1.67602	−0.838011	−	0.545654i	$-0.816281\pi$
$677$	−43.6088	−1.67602	−0.838011	+	0.545654i	$0.816281\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.0684	−0.882686	−0.441343	−	0.897338i	$-0.645498\pi$
$683$	−23.0684	−0.882686	−0.441343	+	0.897338i	$0.645498\pi$
$684$	0	0
$685$	30.7360	1.17436
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−8.85018	−0.336677	−0.168338	−	0.985729i	$-0.553840\pi$
$691$	−8.85018	−0.336677	−0.168338	+	0.985729i	$0.553840\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−76.5993	−2.90558
$696$	0	0
$697$	16.8727	0.639101
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−49.4458	−1.86754	−0.933772	−	0.357869i	$-0.883504\pi$
$701$	−49.4458	−1.86754	−0.933772	+	0.357869i	$0.883504\pi$
$702$	0	0
$703$	−10.4364	−0.393615
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.21819	−0.309077
$708$	0	0
$709$	−12.6658	−0.475676	−0.237838	−	0.971305i	$-0.576439\pi$
$709$	−12.6658	−0.475676	−0.237838	+	0.971305i	$0.576439\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.71345	−0.101620
$714$	0	0
$715$	−5.42690	−0.202955
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.8139	0.701638	0.350819	−	0.936443i	$-0.385903\pi$
$719$	18.8139	0.701638	0.350819	+	0.936443i	$0.385903\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−63.0909	−2.34314
$726$	0	0
$727$	28.1629	1.04451	0.522253	−	0.852791i	$-0.325092\pi$
$727$	28.1629	1.04451	0.522253	+	0.852791i	$0.325092\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.4364	−0.903812
$732$	0	0
$733$	3.84070	0.141860	0.0709298	−	0.997481i	$-0.477403\pi$
$733$	3.84070	0.141860	0.0709298	+	0.997481i	$0.477403\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.8727	−0.621516
$738$	0	0
$739$	8.09275	0.297697	0.148848	−	0.988860i	$-0.452443\pi$
$739$	8.09275	0.297697	0.148848	+	0.988860i	$0.452443\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.9506	−0.988721	−0.494360	−	0.869257i	$-0.664598\pi$
$743$	−26.9506	−0.988721	−0.494360	+	0.869257i	$0.664598\pi$
$744$	0	0
$745$	−21.7229	−0.795866
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.3680	0.488457
$750$	0	0
$751$	−28.4589	−1.03848	−0.519241	−	0.854628i	$-0.673785\pi$
$751$	−28.4589	−1.03848	−0.519241	+	0.854628i	$0.673785\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.4458	−0.416557
$756$	0	0
$757$	−35.6051	−1.29409	−0.647045	−	0.762451i	$-0.723996\pi$
$757$	−35.6051	−1.29409	−0.647045	+	0.762451i	$0.723996\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.35491	−0.0853654	−0.0426827	−	0.999089i	$-0.513590\pi$
$761$	−2.35491	−0.0853654	−0.0426827	+	0.999089i	$0.513590\pi$
$762$	0	0
$763$	6.43637	0.233012
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	31.7229	1.14396	0.571979	−	0.820268i	$-0.306176\pi$
$769$	31.7229	1.14396	0.571979	+	0.820268i	$0.306176\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.42690	−0.0513221	−0.0256611	−	0.999671i	$-0.508169\pi$
$773$	−1.42690	−0.0513221	−0.0256611	+	0.999671i	$0.508169\pi$
$774$	0	0
$775$	17.3567	0.623472
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	−12.4364	−0.445008
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−32.5767	−1.16271
$786$	0	0
$787$	−28.3888	−1.01195	−0.505975	−	0.862548i	$-0.668867\pi$
$787$	−28.3888	−1.01195	−0.505975	+	0.862548i	$0.668867\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.50474	−0.124614
$792$	0	0
$793$	9.28655	0.329775
$794$	0	0
$795$	0	0
$796$	0	0
$797$	53.6051	1.89879	0.949396	−	0.314082i	$-0.101697\pi$
$797$	53.6051	1.89879	0.949396	+	0.314082i	$0.101697\pi$
$798$	0	0
$799$	−23.2200	−0.821465
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.57310	−0.231960
$804$	0	0
$805$	−8.43637	−0.297343
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−54.3186	−1.90974	−0.954870	−	0.297024i	$-0.904006\pi$
$809$	−54.3186	−1.90974	−0.954870	+	0.297024i	$0.904006\pi$
$810$	0	0
$811$	−24.3186	−0.853941	−0.426971	−	0.904266i	$-0.640419\pi$
$811$	−24.3186	−0.853941	−0.426971	+	0.904266i	$0.640419\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19.0095	−0.665873
$816$	0	0
$817$	5.79310	0.202675
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.8822	1.18250	0.591249	−	0.806489i	$-0.298635\pi$
$821$	33.8822	1.18250	0.591249	+	0.806489i	$0.298635\pi$
$822$	0	0
$823$	21.6527	0.754767	0.377384	−	0.926057i	$-0.376824\pi$
$823$	21.6527	0.754767	0.377384	+	0.926057i	$0.376824\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.5178	0.852569	0.426284	−	0.904589i	$-0.359822\pi$
$827$	24.5178	0.852569	0.426284	+	0.904589i	$0.359822\pi$
$828$	0	0
$829$	−54.0451	−1.87707	−0.938533	−	0.345190i	$-0.887814\pi$
$829$	−54.0451	−1.87707	−0.938533	+	0.345190i	$0.887814\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.21819	0.146152
$834$	0	0
$835$	97.6503	3.37933
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.2996	−1.18416	−0.592078	−	0.805881i	$-0.701692\pi$
$839$	−34.2996	−1.18416	−0.592078	+	0.805881i	$0.701692\pi$
$840$	0	0
$841$	−4.67895	−0.161343
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−53.0909	−1.82638
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20.8727	−0.715509
$852$	0	0
$853$	6.14035	0.210242	0.105121	−	0.994459i	$-0.466477\pi$
$853$	6.14035	0.210242	0.105121	+	0.994459i	$0.466477\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.84070	0.199515	0.0997573	−	0.995012i	$-0.468193\pi$
$857$	5.84070	0.199515	0.0997573	+	0.995012i	$0.468193\pi$
$858$	0	0
$859$	43.0320	1.46823	0.734117	−	0.679023i	$-0.237596\pi$
$859$	43.0320	1.46823	0.734117	+	0.679023i	$0.237596\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39.5499	1.34629	0.673147	−	0.739509i	$-0.264942\pi$
$863$	39.5499	1.34629	0.673147	+	0.739509i	$0.264942\pi$
$864$	0	0
$865$	−88.0451	−2.99362
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−30.2996	−1.02785
$870$	0	0
$871$	5.42690	0.183884
$872$	0	0
$873$	0	0
$874$	0	0
$875$	32.8727	1.11130
$876$	0	0
$877$	−36.6182	−1.23651	−0.618255	−	0.785978i	$-0.712160\pi$
$877$	−36.6182	−1.23651	−0.618255	+	0.785978i	$0.712160\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.6640	−0.797261	−0.398631	−	0.917112i	$-0.630514\pi$
$881$	−23.6640	−0.797261	−0.398631	+	0.917112i	$0.630514\pi$
$882$	0	0
$883$	23.4458	0.789015	0.394508	−	0.918893i	$-0.370915\pi$
$883$	23.4458	0.789015	0.394508	+	0.918893i	$0.370915\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0226	0.806599	0.403300	−	0.915068i	$-0.367863\pi$
$887$	24.0226	0.806599	0.403300	+	0.915068i	$0.367863\pi$
$888$	0	0
$889$	−4.43637	−0.148791
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.50474	0.184209
$894$	0	0
$895$	−56.3888	−1.88487
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6.69088	−0.223153
$900$	0	0
$901$	−26.2295	−0.873830
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.7550	−1.22178
$906$	0	0
$907$	−13.8407	−0.459573	−0.229787	−	0.973241i	$-0.573803\pi$
$907$	−13.8407	−0.459573	−0.229787	+	0.973241i	$0.573803\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−19.0909	−0.632511	−0.316255	−	0.948674i	$-0.602426\pi$
$911$	−19.0909	−0.632511	−0.316255	+	0.948674i	$0.602426\pi$
$912$	0	0
$913$	5.86328	0.194046
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19.3680	−0.639588
$918$	0	0
$919$	−23.1724	−0.764387	−0.382193	−	0.924082i	$-0.624831\pi$
$919$	−23.1724	−0.764387	−0.382193	+	0.924082i	$0.624831\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.00000	0.131662
$924$	0	0
$925$	133.514	4.38990
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−35.5084	−1.16499	−0.582496	−	0.812834i	$-0.697924\pi$
$929$	−35.5084	−1.16499	−0.582496	+	0.812834i	$0.697924\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−35.5862	−1.16379
$936$	0	0
$937$	−4.29965	−0.140463	−0.0702317	−	0.997531i	$-0.522374\pi$
$937$	−4.29965	−0.140463	−0.0702317	+	0.997531i	$0.522374\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52.8953	1.72434	0.862169	−	0.506621i	$-0.169106\pi$
$941$	52.8953	1.72434	0.862169	+	0.506621i	$0.169106\pi$
$942$	0	0
$943$	−8.00000	−0.260516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.8858	1.03615	0.518075	−	0.855335i	$-0.326649\pi$
$947$	31.8858	1.03615	0.518075	+	0.855335i	$0.326649\pi$
$948$	0	0
$949$	2.11415	0.0686283
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.2182	−1.23801	−0.619004	−	0.785388i	$-0.712464\pi$
$953$	−38.2182	−1.23801	−0.619004	+	0.785388i	$0.712464\pi$
$954$	0	0
$955$	−3.00947	−0.0973842
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.28655	0.235295
$960$	0	0
$961$	−29.1593	−0.940622
$962$	0	0
$963$	0	0
$964$	0	0
$965$	49.4495	1.59183
$966$	0	0
$967$	11.2200	0.360811	0.180405	−	0.983592i	$-0.442259\pi$
$967$	11.2200	0.360811	0.180405	+	0.983592i	$0.442259\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−56.5993	−1.81636	−0.908179	−	0.418582i	$-0.862527\pi$
$971$	−56.5993	−1.81636	−0.908179	+	0.418582i	$0.862527\pi$
$972$	0	0
$973$	−18.1593	−0.582160
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.6546	0.980726	0.490363	−	0.871518i	$-0.336864\pi$
$977$	30.6546	0.980726	0.490363	+	0.871518i	$0.336864\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29.2865	0.934096	0.467048	−	0.884232i	$-0.345318\pi$
$983$	29.2865	0.934096	0.467048	+	0.884232i	$0.345318\pi$
$984$	0	0
$985$	25.3091	0.806416
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.5862	0.368420
$990$	0	0
$991$	9.01310	0.286311	0.143155	−	0.989700i	$-0.454275\pi$
$991$	9.01310	0.286311	0.143155	+	0.989700i	$0.454275\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	103.077	3.26777
$996$	0	0
$997$	−55.7681	−1.76619	−0.883096	−	0.469192i	$-0.844545\pi$
$997$	−55.7681	−1.76619	−0.883096	+	0.469192i	$0.844545\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.p.1.3		3
3.2	odd	2		1596.2.a.j.1.1	✓	3
12.11	even	2		6384.2.a.bt.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.j.1.1	✓	3	3.2	odd	2
4788.2.a.p.1.3		3	1.1	even	1	trivial
6384.2.a.bt.1.1		3	12.11	even	2

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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+4.21819 q^{5} +1.00000 q^{7} -2.00000 q^{11} +0.643274 q^{13} +4.21819 q^{17} -1.00000 q^{19} -2.00000 q^{23} +12.7931 q^{25} -4.93164 q^{29} +1.35673 q^{31} +4.21819 q^{35} +10.4364 q^{37} +4.00000 q^{41} -5.79310 q^{43} -5.50474 q^{47} +1.00000 q^{49} -6.21819 q^{53} -8.43637 q^{55} +14.4364 q^{61} +2.71345 q^{65} +8.43637 q^{67} +6.21819 q^{71} +3.28655 q^{73} -2.00000 q^{77} +15.1498 q^{79} -2.93164 q^{83} +17.7931 q^{85} +4.00000 q^{89} +0.643274 q^{91} -4.21819 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{7} - 6 q^{11} + 2 q^{13} - 3 q^{19} - 6 q^{23} + 13 q^{25} - 2 q^{29} + 4 q^{31} + 6 q^{37} + 12 q^{41} + 8 q^{43} - 4 q^{47} + 3 q^{49} - 6 q^{53} + 18 q^{61} + 8 q^{65} + 6 q^{71} + 10 q^{73}+ \cdots - 6 q^{97}+O(q^{100}) \) 3 * q + 3 * q^7 - 6 * q^11 + 2 * q^13 - 3 * q^19 - 6 * q^23 + 13 * q^25 - 2 * q^29 + 4 * q^31 + 6 * q^37 + 12 * q^41 + 8 * q^43 - 4 * q^47 + 3 * q^49 - 6 * q^53 + 18 * q^61 + 8 * q^65 + 6 * q^71 + 10 * q^73 - 6 * q^77 + 20 * q^79 + 4 * q^83 + 28 * q^85 + 12 * q^89 + 2 * q^91 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more