LMFDB - Embedded newform 4788.2.a.h.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.41421 q^{5} +1.00000 q^{7} -2.00000 q^{11} +2.82843 q^{13} +4.24264 q^{17} +1.00000 q^{19} -7.65685 q^{23} -3.00000 q^{25} -3.41421 q^{29} +0.828427 q^{31} -1.41421 q^{35} -6.48528 q^{37} -9.65685 q^{41} +2.00000 q^{43} +7.07107 q^{47} +1.00000 q^{49} +7.89949 q^{53} +2.82843 q^{55} -2.34315 q^{59} -4.82843 q^{61} -4.00000 q^{65} +12.4853 q^{67} -12.5858 q^{71} -0.343146 q^{73} -2.00000 q^{77} +14.8284 q^{79} +1.41421 q^{83} -6.00000 q^{85} +4.00000 q^{89} +2.82843 q^{91} -1.41421 q^{95} +7.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 4 q^{11} + 2 q^{19} - 4 q^{23} - 6 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{37} - 8 q^{41} + 4 q^{43} + 2 q^{49} - 4 q^{53} - 16 q^{59} - 4 q^{61} - 8 q^{65} + 8 q^{67} - 28 q^{71} - 12 q^{73}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 4 * q^11 + 2 * q^19 - 4 * q^23 - 6 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^37 - 8 * q^41 + 4 * q^43 + 2 * q^49 - 4 * q^53 - 16 * q^59 - 4 * q^61 - 8 * q^65 + 8 * q^67 - 28 * q^71 - 12 * q^73 - 4 * q^77 + 24 * q^79 - 12 * q^85 + 8 * q^89 + 4 * 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
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	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
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.41421	−0.634004	−0.317002	−	0.948425i	$-0.602676\pi$
$29$	−3.41421	−0.634004	−0.317002	+	0.948425i	$0.602676\pi$
$30$	0	0
$31$	0.828427	0.148790	0.0743950	−	0.997229i	$-0.476297\pi$
$31$	0.828427	0.148790	0.0743950	+	0.997229i	$0.476297\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	−6.48528	−1.06617	−0.533087	−	0.846061i	$-0.678968\pi$
$37$	−6.48528	−1.06617	−0.533087	+	0.846061i	$0.678968\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.65685	−1.50815	−0.754074	−	0.656790i	$-0.771914\pi$
$41$	−9.65685	−1.50815	−0.754074	+	0.656790i	$0.771914\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.07107	1.03142	0.515711	−	0.856763i	$-0.327528\pi$
$47$	7.07107	1.03142	0.515711	+	0.856763i	$0.327528\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.89949	1.08508	0.542540	−	0.840030i	$-0.317463\pi$
$53$	7.89949	1.08508	0.542540	+	0.840030i	$0.317463\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.34315	−0.305052	−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315	−0.305052	−0.152526	+	0.988299i	$0.548741\pi$
$60$	0	0
$61$	−4.82843	−0.618217	−0.309108	−	0.951027i	$-0.600031\pi$
$61$	−4.82843	−0.618217	−0.309108	+	0.951027i	$0.600031\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	12.4853	1.52532	0.762660	−	0.646800i	$-0.223893\pi$
$67$	12.4853	1.52532	0.762660	+	0.646800i	$0.223893\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.5858	−1.49366	−0.746829	−	0.665016i	$-0.768425\pi$
$71$	−12.5858	−1.49366	−0.746829	+	0.665016i	$0.768425\pi$
$72$	0	0
$73$	−0.343146	−0.0401622	−0.0200811	−	0.999798i	$-0.506392\pi$
$73$	−0.343146	−0.0401622	−0.0200811	+	0.999798i	$0.506392\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	14.8284	1.66833	0.834164	−	0.551516i	$-0.185951\pi$
$79$	14.8284	1.66833	0.834164	+	0.551516i	$0.185951\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.41421	0.155230	0.0776151	−	0.996983i	$-0.475269\pi$
$83$	1.41421	0.155230	0.0776151	+	0.996983i	$0.475269\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	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
$91$	2.82843	0.296500
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.41421	−0.145095
$96$	0	0
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−8.24264	−0.820173	−0.410087	−	0.912047i	$-0.634502\pi$
$101$	−8.24264	−0.820173	−0.410087	+	0.912047i	$0.634502\pi$
$102$	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
$107$	−9.75736	−0.943280	−0.471640	−	0.881791i	$-0.656338\pi$
$107$	−9.75736	−0.943280	−0.471640	+	0.881791i	$0.656338\pi$
$108$	0	0
$109$	−2.48528	−0.238047	−0.119023	−	0.992891i	$-0.537976\pi$
$109$	−2.48528	−0.238047	−0.119023	+	0.992891i	$0.537976\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.7279	−1.76177	−0.880887	−	0.473326i	$-0.843053\pi$
$113$	−18.7279	−1.76177	−0.880887	+	0.473326i	$0.843053\pi$
$114$	0	0
$115$	10.8284	1.00976
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.24264	0.388922
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	−19.7990	−1.75688	−0.878438	−	0.477856i	$-0.841414\pi$
$127$	−19.7990	−1.75688	−0.878438	+	0.477856i	$0.841414\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.0711	−1.31677	−0.658383	−	0.752683i	$-0.728759\pi$
$131$	−15.0711	−1.31677	−0.658383	+	0.752683i	$0.728759\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.343146	0.0293169	0.0146585	−	0.999893i	$-0.495334\pi$
$137$	0.343146	0.0293169	0.0146585	+	0.999893i	$0.495334\pi$
$138$	0	0
$139$	−19.3137	−1.63817	−0.819084	−	0.573674i	$-0.805518\pi$
$139$	−19.3137	−1.63817	−0.819084	+	0.573674i	$0.805518\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.65685	−0.473050
$144$	0	0
$145$	4.82843	0.400979
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.51472	−0.451783	−0.225892	−	0.974152i	$-0.572530\pi$
$149$	−5.51472	−0.451783	−0.225892	+	0.974152i	$0.572530\pi$
$150$	0	0
$151$	−19.3137	−1.57173	−0.785864	−	0.618400i	$-0.787781\pi$
$151$	−19.3137	−1.57173	−0.785864	+	0.618400i	$0.787781\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.17157	−0.0941030
$156$	0	0
$157$	−5.51472	−0.440122	−0.220061	−	0.975486i	$-0.570626\pi$
$157$	−5.51472	−0.440122	−0.220061	+	0.975486i	$0.570626\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.65685	−0.603445
$162$	0	0
$163$	15.6569	1.22634	0.613170	−	0.789951i	$-0.289894\pi$
$163$	15.6569	1.22634	0.613170	+	0.789951i	$0.289894\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.48528	−0.656611	−0.328305	−	0.944572i	$-0.606478\pi$
$167$	−8.48528	−0.656611	−0.328305	+	0.944572i	$0.606478\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−9.65685	−0.734197	−0.367099	−	0.930182i	$-0.619649\pi$
$173$	−9.65685	−0.734197	−0.367099	+	0.930182i	$0.619649\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.24264	−0.466597	−0.233298	−	0.972405i	$-0.574952\pi$
$179$	−6.24264	−0.466597	−0.233298	+	0.972405i	$0.574952\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.17157	0.674307
$186$	0	0
$187$	−8.48528	−0.620505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.65685	0.264601	0.132300	−	0.991210i	$-0.457764\pi$
$191$	3.65685	0.264601	0.132300	+	0.991210i	$0.457764\pi$
$192$	0	0
$193$	−5.51472	−0.396958	−0.198479	−	0.980105i	$-0.563600\pi$
$193$	−5.51472	−0.396958	−0.198479	+	0.980105i	$0.563600\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.6569	−1.11550	−0.557752	−	0.830007i	$-0.688336\pi$
$197$	−15.6569	−1.11550	−0.557752	+	0.830007i	$0.688336\pi$
$198$	0	0
$199$	8.48528	0.601506	0.300753	−	0.953702i	$-0.402762\pi$
$199$	8.48528	0.601506	0.300753	+	0.953702i	$0.402762\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.41421	−0.239631
$204$	0	0
$205$	13.6569	0.953836
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	21.6569	1.49092	0.745460	−	0.666551i	$-0.232230\pi$
$211$	21.6569	1.49092	0.745460	+	0.666551i	$0.232230\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.82843	−0.192897
$216$	0	0
$217$	0.828427	0.0562373
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−14.4853	−0.970006	−0.485003	−	0.874512i	$-0.661182\pi$
$223$	−14.4853	−0.970006	−0.485003	+	0.874512i	$0.661182\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.17157	0.0777600	0.0388800	−	0.999244i	$-0.487621\pi$
$227$	1.17157	0.0777600	0.0388800	+	0.999244i	$0.487621\pi$
$228$	0	0
$229$	−13.7990	−0.911863	−0.455931	−	0.890015i	$-0.650694\pi$
$229$	−13.7990	−0.911863	−0.455931	+	0.890015i	$0.650694\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.1421	1.05751	0.528753	−	0.848776i	$-0.322660\pi$
$233$	16.1421	1.05751	0.528753	+	0.848776i	$0.322660\pi$
$234$	0	0
$235$	−10.0000	−0.652328
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−25.7990	−1.66880	−0.834399	−	0.551161i	$-0.814185\pi$
$239$	−25.7990	−1.66880	−0.834399	+	0.551161i	$0.814185\pi$
$240$	0	0
$241$	−23.6569	−1.52387	−0.761936	−	0.647652i	$-0.775751\pi$
$241$	−23.6569	−1.52387	−0.761936	+	0.647652i	$0.775751\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.41421	−0.0903508
$246$	0	0
$247$	2.82843	0.179969
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.75736	0.489640	0.244820	−	0.969569i	$-0.421271\pi$
$251$	7.75736	0.489640	0.244820	+	0.969569i	$0.421271\pi$
$252$	0	0
$253$	15.3137	0.962765
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.48528	0.529297	0.264649	−	0.964345i	$-0.414744\pi$
$257$	8.48528	0.529297	0.264649	+	0.964345i	$0.414744\pi$
$258$	0	0
$259$	−6.48528	−0.402976
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.8284	1.03769	0.518843	−	0.854870i	$-0.326363\pi$
$263$	16.8284	1.03769	0.518843	+	0.854870i	$0.326363\pi$
$264$	0	0
$265$	−11.1716	−0.686264
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.51472	−0.458180	−0.229090	−	0.973405i	$-0.573575\pi$
$269$	−7.51472	−0.458180	−0.229090	+	0.973405i	$0.573575\pi$
$270$	0	0
$271$	−23.7990	−1.44569	−0.722843	−	0.691012i	$-0.757165\pi$
$271$	−23.7990	−1.44569	−0.722843	+	0.691012i	$0.757165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	4.97056	0.298652	0.149326	−	0.988788i	$-0.452290\pi$
$277$	4.97056	0.298652	0.149326	+	0.988788i	$0.452290\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.0416	1.55351	0.776757	−	0.629801i	$-0.216863\pi$
$281$	26.0416	1.55351	0.776757	+	0.629801i	$0.216863\pi$
$282$	0	0
$283$	−31.1127	−1.84946	−0.924729	−	0.380626i	$-0.875708\pi$
$283$	−31.1127	−1.84946	−0.924729	+	0.380626i	$0.875708\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.65685	−0.570026
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	0	0
$295$	3.31371	0.192932
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−21.6569	−1.25245
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.82843	0.390995
$306$	0	0
$307$	−20.8284	−1.18874	−0.594371	−	0.804191i	$-0.702599\pi$
$307$	−20.8284	−1.18874	−0.594371	+	0.804191i	$0.702599\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.2132	1.20289	0.601445	−	0.798914i	$-0.294592\pi$
$311$	21.2132	1.20289	0.601445	+	0.798914i	$0.294592\pi$
$312$	0	0
$313$	4.14214	0.234127	0.117064	−	0.993124i	$-0.462652\pi$
$313$	4.14214	0.234127	0.117064	+	0.993124i	$0.462652\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.72792	−0.153215	−0.0766077	−	0.997061i	$-0.524409\pi$
$317$	−2.72792	−0.153215	−0.0766077	+	0.997061i	$0.524409\pi$
$318$	0	0
$319$	6.82843	0.382319
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.24264	0.236067
$324$	0	0
$325$	−8.48528	−0.470679
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.07107	0.389841
$330$	0	0
$331$	19.3137	1.06158	0.530789	−	0.847504i	$-0.321896\pi$
$331$	19.3137	1.06158	0.530789	+	0.847504i	$0.321896\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.6569	−0.964697
$336$	0	0
$337$	−28.1421	−1.53300	−0.766500	−	0.642244i	$-0.778003\pi$
$337$	−28.1421	−1.53300	−0.766500	+	0.642244i	$0.778003\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.65685	−0.0897237
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.4853	1.63654	0.818268	−	0.574837i	$-0.194935\pi$
$347$	30.4853	1.63654	0.818268	+	0.574837i	$0.194935\pi$
$348$	0	0
$349$	10.9706	0.587241	0.293620	−	0.955922i	$-0.405140\pi$
$349$	10.9706	0.587241	0.293620	+	0.955922i	$0.405140\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	0	0
$355$	17.7990	0.944672
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.485281	0.0254008
$366$	0	0
$367$	−11.5147	−0.601063	−0.300532	−	0.953772i	$-0.597164\pi$
$367$	−11.5147	−0.601063	−0.300532	+	0.953772i	$0.597164\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.89949	0.410121
$372$	0	0
$373$	14.9706	0.775146	0.387573	−	0.921839i	$-0.373313\pi$
$373$	14.9706	0.775146	0.387573	+	0.921839i	$0.373313\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.65685	−0.497353
$378$	0	0
$379$	32.2843	1.65833	0.829166	−	0.559003i	$-0.188816\pi$
$379$	32.2843	1.65833	0.829166	+	0.559003i	$0.188816\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10.8284	0.553307	0.276653	−	0.960970i	$-0.410775\pi$
$383$	10.8284	0.553307	0.276653	+	0.960970i	$0.410775\pi$
$384$	0	0
$385$	2.82843	0.144150
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.14214	−0.210015	−0.105007	−	0.994471i	$-0.533487\pi$
$389$	−4.14214	−0.210015	−0.105007	+	0.994471i	$0.533487\pi$
$390$	0	0
$391$	−32.4853	−1.64285
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−20.9706	−1.05514
$396$	0	0
$397$	−13.3137	−0.668196	−0.334098	−	0.942538i	$-0.608432\pi$
$397$	−13.3137	−0.668196	−0.334098	+	0.942538i	$0.608432\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−31.4142	−1.56875	−0.784375	−	0.620286i	$-0.787016\pi$
$401$	−31.4142	−1.56875	−0.784375	+	0.620286i	$0.787016\pi$
$402$	0	0
$403$	2.34315	0.116720
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12.9706	0.642927
$408$	0	0
$409$	−14.8284	−0.733219	−0.366609	−	0.930375i	$-0.619481\pi$
$409$	−14.8284	−0.733219	−0.366609	+	0.930375i	$0.619481\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.34315	−0.115299
$414$	0	0
$415$	−2.00000	−0.0981761
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.242641	0.0118538	0.00592689	−	0.999982i	$-0.498113\pi$
$419$	0.242641	0.0118538	0.00592689	+	0.999982i	$0.498113\pi$
$420$	0	0
$421$	−22.2843	−1.08607	−0.543034	−	0.839710i	$-0.682725\pi$
$421$	−22.2843	−1.08607	−0.543034	+	0.839710i	$0.682725\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.7279	−0.617395
$426$	0	0
$427$	−4.82843	−0.233664
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.5858	1.37693	0.688464	−	0.725270i	$-0.258285\pi$
$431$	28.5858	1.37693	0.688464	+	0.725270i	$0.258285\pi$
$432$	0	0
$433$	−33.3137	−1.60095	−0.800477	−	0.599363i	$-0.795421\pi$
$433$	−33.3137	−1.60095	−0.800477	+	0.599363i	$0.795421\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.65685	−0.366277
$438$	0	0
$439$	9.51472	0.454113	0.227056	−	0.973882i	$-0.427090\pi$
$439$	9.51472	0.454113	0.227056	+	0.973882i	$0.427090\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−25.7990	−1.22575	−0.612873	−	0.790181i	$-0.709986\pi$
$443$	−25.7990	−1.22575	−0.612873	+	0.790181i	$0.709986\pi$
$444$	0	0
$445$	−5.65685	−0.268161
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.75736	−0.271707	−0.135853	−	0.990729i	$-0.543378\pi$
$449$	−5.75736	−0.271707	−0.135853	+	0.990729i	$0.543378\pi$
$450$	0	0
$451$	19.3137	0.909447
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	−3.31371	−0.155009	−0.0775044	−	0.996992i	$-0.524695\pi$
$457$	−3.31371	−0.155009	−0.0775044	+	0.996992i	$0.524695\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.0711	0.888228	0.444114	−	0.895970i	$-0.353518\pi$
$461$	19.0711	0.888228	0.444114	+	0.895970i	$0.353518\pi$
$462$	0	0
$463$	−15.3137	−0.711688	−0.355844	−	0.934545i	$-0.615807\pi$
$463$	−15.3137	−0.711688	−0.355844	+	0.934545i	$0.615807\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	38.8701	1.79869	0.899346	−	0.437238i	$-0.144043\pi$
$467$	38.8701	1.79869	0.899346	+	0.437238i	$0.144043\pi$
$468$	0	0
$469$	12.4853	0.576517
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	−3.00000	−0.137649
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.0711	−0.688615	−0.344307	−	0.938857i	$-0.611886\pi$
$479$	−15.0711	−0.688615	−0.344307	+	0.938857i	$0.611886\pi$
$480$	0	0
$481$	−18.3431	−0.836375
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.8284	−0.491694
$486$	0	0
$487$	−20.2843	−0.919168	−0.459584	−	0.888134i	$-0.652001\pi$
$487$	−20.2843	−0.919168	−0.459584	+	0.888134i	$0.652001\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.82843	−0.217904	−0.108952	−	0.994047i	$-0.534749\pi$
$491$	−4.82843	−0.217904	−0.108952	+	0.994047i	$0.534749\pi$
$492$	0	0
$493$	−14.4853	−0.652384
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.5858	−0.564550
$498$	0	0
$499$	15.3137	0.685536	0.342768	−	0.939420i	$-0.388636\pi$
$499$	15.3137	0.685536	0.342768	+	0.939420i	$0.388636\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	32.2426	1.43763	0.718814	−	0.695202i	$-0.244685\pi$
$503$	32.2426	1.43763	0.718814	+	0.695202i	$0.244685\pi$
$504$	0	0
$505$	11.6569	0.518723
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−31.1127	−1.37905	−0.689523	−	0.724264i	$-0.742180\pi$
$509$	−31.1127	−1.37905	−0.689523	+	0.724264i	$0.742180\pi$
$510$	0	0
$511$	−0.343146	−0.0151799
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.3137	−0.498542
$516$	0	0
$517$	−14.1421	−0.621970
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.45584	−0.414268	−0.207134	−	0.978313i	$-0.566414\pi$
$521$	−9.45584	−0.414268	−0.207134	+	0.978313i	$0.566414\pi$
$522$	0	0
$523$	33.7990	1.47793	0.738963	−	0.673746i	$-0.235316\pi$
$523$	33.7990	1.47793	0.738963	+	0.673746i	$0.235316\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.51472	0.153104
$528$	0	0
$529$	35.6274	1.54902
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−27.3137	−1.18309
$534$	0	0
$535$	13.7990	0.596582
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	17.3137	0.744374	0.372187	−	0.928158i	$-0.378608\pi$
$541$	17.3137	0.744374	0.372187	+	0.928158i	$0.378608\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.51472	0.150554
$546$	0	0
$547$	7.51472	0.321306	0.160653	−	0.987011i	$-0.448640\pi$
$547$	7.51472	0.321306	0.160653	+	0.987011i	$0.448640\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.41421	−0.145450
$552$	0	0
$553$	14.8284	0.630569
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.7990	1.26262	0.631312	−	0.775529i	$-0.282517\pi$
$557$	29.7990	1.26262	0.631312	+	0.775529i	$0.282517\pi$
$558$	0	0
$559$	5.65685	0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.85786	−0.246880	−0.123440	−	0.992352i	$-0.539393\pi$
$563$	−5.85786	−0.246880	−0.123440	+	0.992352i	$0.539393\pi$
$564$	0	0
$565$	26.4853	1.11424
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.4142	0.813886	0.406943	−	0.913454i	$-0.366595\pi$
$569$	19.4142	0.813886	0.406943	+	0.913454i	$0.366595\pi$
$570$	0	0
$571$	36.2843	1.51845	0.759225	−	0.650829i	$-0.225578\pi$
$571$	36.2843	1.51845	0.759225	+	0.650829i	$0.225578\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	22.9706	0.957939
$576$	0	0
$577$	−10.4853	−0.436508	−0.218254	−	0.975892i	$-0.570036\pi$
$577$	−10.4853	−0.436508	−0.218254	+	0.975892i	$0.570036\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.41421	0.0586715
$582$	0	0
$583$	−15.7990	−0.654327
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.0711	1.11734	0.558671	−	0.829389i	$-0.311311\pi$
$587$	27.0711	1.11734	0.558671	+	0.829389i	$0.311311\pi$
$588$	0	0
$589$	0.828427	0.0341347
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10.8701	0.446380	0.223190	−	0.974775i	$-0.428353\pi$
$593$	10.8701	0.446380	0.223190	+	0.974775i	$0.428353\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.72792	0.111460	0.0557299	−	0.998446i	$-0.482251\pi$
$599$	2.72792	0.111460	0.0557299	+	0.998446i	$0.482251\pi$
$600$	0	0
$601$	−21.3137	−0.869404	−0.434702	−	0.900574i	$-0.643146\pi$
$601$	−21.3137	−0.869404	−0.434702	+	0.900574i	$0.643146\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.89949	0.402472
$606$	0	0
$607$	22.6274	0.918419	0.459209	−	0.888328i	$-0.348133\pi$
$607$	22.6274	0.918419	0.459209	+	0.888328i	$0.348133\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20.0000	0.809113
$612$	0	0
$613$	20.2843	0.819274	0.409637	−	0.912249i	$-0.365655\pi$
$613$	20.2843	0.819274	0.409637	+	0.912249i	$0.365655\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9.31371	−0.374956	−0.187478	−	0.982269i	$-0.560031\pi$
$617$	−9.31371	−0.374956	−0.187478	+	0.982269i	$0.560031\pi$
$618$	0	0
$619$	20.6863	0.831452	0.415726	−	0.909490i	$-0.363527\pi$
$619$	20.6863	0.831452	0.415726	+	0.909490i	$0.363527\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−27.5147	−1.09708
$630$	0	0
$631$	40.6274	1.61735	0.808676	−	0.588254i	$-0.200185\pi$
$631$	40.6274	1.61735	0.808676	+	0.588254i	$0.200185\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	28.0000	1.11115
$636$	0	0
$637$	2.82843	0.112066
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.92893	−0.115686	−0.0578429	−	0.998326i	$-0.518422\pi$
$641$	−2.92893	−0.115686	−0.0578429	+	0.998326i	$0.518422\pi$
$642$	0	0
$643$	−7.51472	−0.296352	−0.148176	−	0.988961i	$-0.547340\pi$
$643$	−7.51472	−0.296352	−0.148176	+	0.988961i	$0.547340\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.9289	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$647$	−24.9289	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$648$	0	0
$649$	4.68629	0.183953
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.9411	1.40648	0.703242	−	0.710950i	$-0.251735\pi$
$653$	35.9411	1.40648	0.703242	+	0.710950i	$0.251735\pi$
$654$	0	0
$655$	21.3137	0.832796
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.27208	0.0495531	0.0247766	−	0.999693i	$-0.492113\pi$
$659$	1.27208	0.0495531	0.0247766	+	0.999693i	$0.492113\pi$
$660$	0	0
$661$	−11.7990	−0.458928	−0.229464	−	0.973317i	$-0.573697\pi$
$661$	−11.7990	−0.458928	−0.229464	+	0.973317i	$0.573697\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.41421	−0.0548408
$666$	0	0
$667$	26.1421	1.01223
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.65685	0.372799
$672$	0	0
$673$	9.51472	0.366765	0.183383	−	0.983042i	$-0.441295\pi$
$673$	9.51472	0.366765	0.183383	+	0.983042i	$0.441295\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−35.1127	−1.34949	−0.674745	−	0.738051i	$-0.735747\pi$
$677$	−35.1127	−1.34949	−0.674745	+	0.738051i	$0.735747\pi$
$678$	0	0
$679$	7.65685	0.293843
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8.78680	0.336217	0.168109	−	0.985768i	$-0.446234\pi$
$683$	8.78680	0.336217	0.168109	+	0.985768i	$0.446234\pi$
$684$	0	0
$685$	−0.485281	−0.0185416
$686$	0	0
$687$	0	0
$688$	0	0
$689$	22.3431	0.851206
$690$	0	0
$691$	−19.7990	−0.753189	−0.376595	−	0.926378i	$-0.622905\pi$
$691$	−19.7990	−0.753189	−0.376595	+	0.926378i	$0.622905\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.3137	1.03607
$696$	0	0
$697$	−40.9706	−1.55187
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25.3137	−0.956086	−0.478043	−	0.878337i	$-0.658654\pi$
$701$	−25.3137	−0.956086	−0.478043	+	0.878337i	$0.658654\pi$
$702$	0	0
$703$	−6.48528	−0.244597
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.24264	−0.309996
$708$	0	0
$709$	−35.5980	−1.33691	−0.668455	−	0.743753i	$-0.733044\pi$
$709$	−35.5980	−1.33691	−0.668455	+	0.743753i	$0.733044\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.34315	−0.237553
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.5858	−0.394783	−0.197392	−	0.980325i	$-0.563247\pi$
$719$	−10.5858	−0.394783	−0.197392	+	0.980325i	$0.563247\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.2426	0.380402
$726$	0	0
$727$	17.4558	0.647401	0.323701	−	0.946160i	$-0.395073\pi$
$727$	17.4558	0.647401	0.323701	+	0.946160i	$0.395073\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.48528	0.313839
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.9706	−0.919803
$738$	0	0
$739$	12.3431	0.454050	0.227025	−	0.973889i	$-0.427100\pi$
$739$	12.3431	0.454050	0.227025	+	0.973889i	$0.427100\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.7574	−0.798200	−0.399100	−	0.916907i	$-0.630677\pi$
$743$	−21.7574	−0.798200	−0.399100	+	0.916907i	$0.630677\pi$
$744$	0	0
$745$	7.79899	0.285733
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.75736	−0.356526
$750$	0	0
$751$	−22.6274	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$751$	−22.6274	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	27.3137	0.994048
$756$	0	0
$757$	−6.97056	−0.253349	−0.126675	−	0.991944i	$-0.540430\pi$
$757$	−6.97056	−0.253349	−0.126675	+	0.991944i	$0.540430\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.41421	−0.196265	−0.0981325	−	0.995173i	$-0.531287\pi$
$761$	−5.41421	−0.196265	−0.0981325	+	0.995173i	$0.531287\pi$
$762$	0	0
$763$	−2.48528	−0.0899732
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.62742	−0.239302
$768$	0	0
$769$	−49.1127	−1.77105	−0.885525	−	0.464592i	$-0.846201\pi$
$769$	−49.1127	−1.77105	−0.885525	+	0.464592i	$0.846201\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.6274	1.24546	0.622731	−	0.782436i	$-0.286023\pi$
$773$	34.6274	1.24546	0.622731	+	0.782436i	$0.286023\pi$
$774$	0	0
$775$	−2.48528	−0.0892739
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.65685	−0.345993
$780$	0	0
$781$	25.1716	0.900710
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.79899	0.278358
$786$	0	0
$787$	8.14214	0.290236	0.145118	−	0.989414i	$-0.453644\pi$
$787$	8.14214	0.290236	0.145118	+	0.989414i	$0.453644\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−18.7279	−0.665888
$792$	0	0
$793$	−13.6569	−0.484969
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.9411	1.20226	0.601128	−	0.799153i	$-0.294718\pi$
$797$	33.9411	1.20226	0.601128	+	0.799153i	$0.294718\pi$
$798$	0	0
$799$	30.0000	1.06132
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.686292	0.0242187
$804$	0	0
$805$	10.8284	0.381652
$806$	0	0
$807$	0	0
$808$	0	0
$809$	35.6569	1.25363	0.626814	−	0.779169i	$-0.284359\pi$
$809$	35.6569	1.25363	0.626814	+	0.779169i	$0.284359\pi$
$810$	0	0
$811$	−34.6274	−1.21593	−0.607967	−	0.793963i	$-0.708015\pi$
$811$	−34.6274	−1.21593	−0.607967	+	0.793963i	$0.708015\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−22.1421	−0.775605
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.4558	−1.79582	−0.897911	−	0.440178i	$-0.854915\pi$
$821$	−51.4558	−1.79582	−0.897911	+	0.440178i	$0.854915\pi$
$822$	0	0
$823$	−27.6569	−0.964057	−0.482029	−	0.876155i	$-0.660100\pi$
$823$	−27.6569	−0.964057	−0.482029	+	0.876155i	$0.660100\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.75736	0.0611094	0.0305547	−	0.999533i	$-0.490273\pi$
$827$	1.75736	0.0611094	0.0305547	+	0.999533i	$0.490273\pi$
$828$	0	0
$829$	55.9411	1.94292	0.971458	−	0.237212i	$-0.0762338\pi$
$829$	55.9411	1.94292	0.971458	+	0.237212i	$0.0762338\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.24264	0.146999
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−12.9706	−0.447794	−0.223897	−	0.974613i	$-0.571878\pi$
$839$	−12.9706	−0.447794	−0.223897	+	0.974613i	$0.571878\pi$
$840$	0	0
$841$	−17.3431	−0.598040
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.07107	0.243252
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	49.6569	1.70222
$852$	0	0
$853$	30.2843	1.03691	0.518457	−	0.855104i	$-0.326507\pi$
$853$	30.2843	1.03691	0.518457	+	0.855104i	$0.326507\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.3137	−0.386469	−0.193234	−	0.981153i	$-0.561898\pi$
$857$	−11.3137	−0.386469	−0.193234	+	0.981153i	$0.561898\pi$
$858$	0	0
$859$	−50.9117	−1.73708	−0.868542	−	0.495615i	$-0.834943\pi$
$859$	−50.9117	−1.73708	−0.868542	+	0.495615i	$0.834943\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.7574	0.740629	0.370315	−	0.928906i	$-0.379250\pi$
$863$	21.7574	0.740629	0.370315	+	0.928906i	$0.379250\pi$
$864$	0	0
$865$	13.6569	0.464347
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−29.6569	−1.00604
$870$	0	0
$871$	35.3137	1.19656
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.3137	0.382473
$876$	0	0
$877$	−9.02944	−0.304902	−0.152451	−	0.988311i	$-0.548717\pi$
$877$	−9.02944	−0.304902	−0.152451	+	0.988311i	$0.548717\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−54.1838	−1.82550	−0.912749	−	0.408521i	$-0.866045\pi$
$881$	−54.1838	−1.82550	−0.912749	+	0.408521i	$0.866045\pi$
$882$	0	0
$883$	−56.2843	−1.89412	−0.947058	−	0.321062i	$-0.895960\pi$
$883$	−56.2843	−1.89412	−0.947058	+	0.321062i	$0.895960\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.8284	0.363583	0.181791	−	0.983337i	$-0.441810\pi$
$887$	10.8284	0.363583	0.181791	+	0.983337i	$0.441810\pi$
$888$	0	0
$889$	−19.7990	−0.664037
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7.07107	0.236624
$894$	0	0
$895$	8.82843	0.295102
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.82843	−0.0943333
$900$	0	0
$901$	33.5147	1.11654
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.48528	0.282060
$906$	0	0
$907$	38.3431	1.27316	0.636582	−	0.771209i	$-0.280348\pi$
$907$	38.3431	1.27316	0.636582	+	0.771209i	$0.280348\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	41.5563	1.37682	0.688412	−	0.725320i	$-0.258308\pi$
$911$	41.5563	1.37682	0.688412	+	0.725320i	$0.258308\pi$
$912$	0	0
$913$	−2.82843	−0.0936073
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.0711	−0.497691
$918$	0	0
$919$	30.6274	1.01031	0.505153	−	0.863030i	$-0.331436\pi$
$919$	30.6274	1.01031	0.505153	+	0.863030i	$0.331436\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−35.5980	−1.17172
$924$	0	0
$925$	19.4558	0.639704
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13.8995	0.456028	0.228014	−	0.973658i	$-0.426777\pi$
$929$	13.8995	0.456028	0.228014	+	0.973658i	$0.426777\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	−1.71573	−0.0560504	−0.0280252	−	0.999607i	$-0.508922\pi$
$937$	−1.71573	−0.0560504	−0.0280252	+	0.999607i	$0.508922\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.1421	−0.852209	−0.426105	−	0.904674i	$-0.640114\pi$
$941$	−26.1421	−0.852209	−0.426105	+	0.904674i	$0.640114\pi$
$942$	0	0
$943$	73.9411	2.40785
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.6274	0.800284	0.400142	−	0.916453i	$-0.368961\pi$
$947$	24.6274	0.800284	0.400142	+	0.916453i	$0.368961\pi$
$948$	0	0
$949$	−0.970563	−0.0315058
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−29.3553	−0.950913	−0.475456	−	0.879739i	$-0.657717\pi$
$953$	−29.3553	−0.950913	−0.475456	+	0.879739i	$0.657717\pi$
$954$	0	0
$955$	−5.17157	−0.167348
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.343146	0.0110808
$960$	0	0
$961$	−30.3137	−0.977862
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.79899	0.251058
$966$	0	0
$967$	4.62742	0.148808	0.0744038	−	0.997228i	$-0.476295\pi$
$967$	4.62742	0.148808	0.0744038	+	0.997228i	$0.476295\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	59.5980	1.91259	0.956295	−	0.292403i	$-0.0944550\pi$
$971$	59.5980	1.91259	0.956295	+	0.292403i	$0.0944550\pi$
$972$	0	0
$973$	−19.3137	−0.619169
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.24264	0.199720	0.0998599	−	0.995002i	$-0.468161\pi$
$977$	6.24264	0.199720	0.0998599	+	0.995002i	$0.468161\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20.0000	−0.637901	−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000	−0.637901	−0.318950	+	0.947771i	$0.603330\pi$
$984$	0	0
$985$	22.1421	0.705507
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−15.3137	−0.486948
$990$	0	0
$991$	−55.5980	−1.76613	−0.883064	−	0.469253i	$-0.844523\pi$
$991$	−55.5980	−1.76613	−0.883064	+	0.469253i	$0.844523\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−25.7990	−0.817062	−0.408531	−	0.912744i	$-0.633959\pi$
$997$	−25.7990	−0.817062	−0.408531	+	0.912744i	$0.633959\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.h.1.1		2
3.2	odd	2		1596.2.a.h.1.2	✓	2
12.11	even	2		6384.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.h.1.2	✓	2	3.2	odd	2
4788.2.a.h.1.1		2	1.1	even	1	trivial
6384.2.a.bp.1.2		2	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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{5} +1.00000 q^{7} -2.00000 q^{11} +2.82843 q^{13} +4.24264 q^{17} +1.00000 q^{19} -7.65685 q^{23} -3.00000 q^{25} -3.41421 q^{29} +0.828427 q^{31} -1.41421 q^{35} -6.48528 q^{37} -9.65685 q^{41} +2.00000 q^{43} +7.07107 q^{47} +1.00000 q^{49} +7.89949 q^{53} +2.82843 q^{55} -2.34315 q^{59} -4.82843 q^{61} -4.00000 q^{65} +12.4853 q^{67} -12.5858 q^{71} -0.343146 q^{73} -2.00000 q^{77} +14.8284 q^{79} +1.41421 q^{83} -6.00000 q^{85} +4.00000 q^{89} +2.82843 q^{91} -1.41421 q^{95} +7.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 4 q^{11} + 2 q^{19} - 4 q^{23} - 6 q^{25} - 4 q^{29} - 4 q^{31} + 4 q^{37} - 8 q^{41} + 4 q^{43} + 2 q^{49} - 4 q^{53} - 16 q^{59} - 4 q^{61} - 8 q^{65} + 8 q^{67} - 28 q^{71} - 12 q^{73}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 4 * q^11 + 2 * q^19 - 4 * q^23 - 6 * q^25 - 4 * q^29 - 4 * q^31 + 4 * q^37 - 8 * q^41 + 4 * q^43 + 2 * q^49 - 4 * q^53 - 16 * q^59 - 4 * q^61 - 8 * q^65 + 8 * q^67 - 28 * q^71 - 12 * q^73 - 4 * q^77 + 24 * q^79 - 12 * q^85 + 8 * q^89 + 4 * q^97

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