LMFDB - Embedded newform 4788.2.a.s.1.4

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:	$4$
Coefficient field:	4.4.32448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} + 12 $ x^4 - 10*x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.93352$ 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)

$f(q)$	$=$	$q+3.82172 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+3.82172 q^{5} +1.00000 q^{7} +5.86704 q^{11} -0.605551 q^{13} -2.04532 q^{17} +1.00000 q^{19} +5.86704 q^{23} +9.60555 q^{25} -2.04532 q^{29} -6.60555 q^{31} +3.82172 q^{35} +2.00000 q^{37} +5.86704 q^{41} -6.60555 q^{43} +2.04532 q^{47} +1.00000 q^{49} +3.82172 q^{53} +22.4222 q^{55} +11.7341 q^{59} +2.00000 q^{61} -2.31425 q^{65} -9.21110 q^{67} -15.5558 q^{71} +7.21110 q^{73} +5.86704 q^{77} -4.00000 q^{79} -3.82172 q^{83} -7.81665 q^{85} -9.41984 q^{89} -0.605551 q^{91} +3.82172 q^{95} -3.21110 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 12 q^{13} + 4 q^{19} + 24 q^{25} - 12 q^{31} + 8 q^{37} - 12 q^{43} + 4 q^{49} + 32 q^{55} + 8 q^{61} - 8 q^{67} - 16 q^{79} + 12 q^{85} + 12 q^{91} + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 12 * q^13 + 4 * q^19 + 24 * q^25 - 12 * q^31 + 8 * q^37 - 12 * q^43 + 4 * q^49 + 32 * q^55 + 8 * q^61 - 8 * q^67 - 16 * q^79 + 12 * q^85 + 12 * q^91 + 16 * 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
$5$	3.82172	1.70913	0.854563	−	0.519348i	$-0.173825\pi$
$5$	3.82172	1.70913	0.854563	+	0.519348i	$0.173825\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.86704	1.76898	0.884490	−	0.466559i	$-0.154506\pi$
$11$	5.86704	1.76898	0.884490	+	0.466559i	$0.154506\pi$
$12$	0	0
$13$	−0.605551	−0.167950	−0.0839749	−	0.996468i	$-0.526762\pi$
$13$	−0.605551	−0.167950	−0.0839749	+	0.996468i	$0.526762\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.04532	−0.496064	−0.248032	−	0.968752i	$-0.579784\pi$
$17$	−2.04532	−0.496064	−0.248032	+	0.968752i	$0.579784\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.86704	1.22336	0.611682	−	0.791104i	$-0.290493\pi$
$23$	5.86704	1.22336	0.611682	+	0.791104i	$0.290493\pi$
$24$	0	0
$25$	9.60555	1.92111
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.04532	−0.379807	−0.189903	−	0.981803i	$-0.560818\pi$
$29$	−2.04532	−0.379807	−0.189903	+	0.981803i	$0.560818\pi$
$30$	0	0
$31$	−6.60555	−1.18639	−0.593196	−	0.805058i	$-0.702134\pi$
$31$	−6.60555	−1.18639	−0.593196	+	0.805058i	$0.702134\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.82172	0.645989
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.86704	0.916278	0.458139	−	0.888880i	$-0.348516\pi$
$41$	5.86704	0.916278	0.458139	+	0.888880i	$0.348516\pi$
$42$	0	0
$43$	−6.60555	−1.00734	−0.503669	−	0.863897i	$-0.668017\pi$
$43$	−6.60555	−1.00734	−0.503669	+	0.863897i	$0.668017\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.04532	0.298341	0.149171	−	0.988811i	$-0.452340\pi$
$47$	2.04532	0.298341	0.149171	+	0.988811i	$0.452340\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.82172	0.524954	0.262477	−	0.964938i	$-0.415461\pi$
$53$	3.82172	0.524954	0.262477	+	0.964938i	$0.415461\pi$
$54$	0	0
$55$	22.4222	3.02341
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.7341	1.52765	0.763824	−	0.645425i	$-0.223320\pi$
$59$	11.7341	1.52765	0.763824	+	0.645425i	$0.223320\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.31425	−0.287047
$66$	0	0
$67$	−9.21110	−1.12532	−0.562658	−	0.826690i	$-0.690221\pi$
$67$	−9.21110	−1.12532	−0.562658	+	0.826690i	$0.690221\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.5558	−1.84613	−0.923067	−	0.384638i	$-0.874326\pi$
$71$	−15.5558	−1.84613	−0.923067	+	0.384638i	$0.874326\pi$
$72$	0	0
$73$	7.21110	0.843996	0.421998	−	0.906597i	$-0.361329\pi$
$73$	7.21110	0.843996	0.421998	+	0.906597i	$0.361329\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.86704	0.668612
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.82172	−0.419488	−0.209744	−	0.977756i	$-0.567263\pi$
$83$	−3.82172	−0.419488	−0.209744	+	0.977756i	$0.567263\pi$
$84$	0	0
$85$	−7.81665	−0.847835
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.41984	−0.998501	−0.499251	−	0.866458i	$-0.666391\pi$
$89$	−9.41984	−0.998501	−0.499251	+	0.866458i	$0.666391\pi$
$90$	0	0
$91$	−0.605551	−0.0634790
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.82172	0.392100
$96$	0	0
$97$	−3.21110	−0.326038	−0.163019	−	0.986623i	$-0.552123\pi$
$97$	−3.21110	−0.326038	−0.163019	+	0.986623i	$0.552123\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.13597	−0.610552	−0.305276	−	0.952264i	$-0.598749\pi$
$101$	−6.13597	−0.610552	−0.305276	+	0.952264i	$0.598749\pi$
$102$	0	0
$103$	18.4222	1.81519	0.907597	−	0.419843i	$-0.137915\pi$
$103$	18.4222	1.81519	0.907597	+	0.419843i	$0.137915\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.04532	−0.197729	−0.0988644	−	0.995101i	$-0.531521\pi$
$107$	−2.04532	−0.197729	−0.0988644	+	0.995101i	$0.531521\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.7794	−1.29626	−0.648129	−	0.761531i	$-0.724448\pi$
$113$	−13.7794	−1.29626	−0.648129	+	0.761531i	$0.724448\pi$
$114$	0	0
$115$	22.4222	2.09088
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.04532	−0.187494
$120$	0	0
$121$	23.4222	2.12929
$122$	0	0
$123$	0	0
$124$	0	0
$125$	17.6011	1.57429
$126$	0	0
$127$	13.2111	1.17230	0.586148	−	0.810204i	$-0.300644\pi$
$127$	13.2111	1.17230	0.586148	+	0.810204i	$0.300644\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.4652	−1.00172	−0.500858	−	0.865529i	$-0.666982\pi$
$131$	−11.4652	−1.00172	−0.500858	+	0.865529i	$0.666982\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	2.78890	0.236551	0.118276	−	0.992981i	$-0.462263\pi$
$139$	2.78890	0.236551	0.118276	+	0.992981i	$0.462263\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.55280	−0.297100
$144$	0	0
$145$	−7.81665	−0.649138
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.64344	0.626175	0.313088	−	0.949724i	$-0.398637\pi$
$149$	7.64344	0.626175	0.313088	+	0.949724i	$0.398637\pi$
$150$	0	0
$151$	−14.4222	−1.17366	−0.586831	−	0.809709i	$-0.699625\pi$
$151$	−14.4222	−1.17366	−0.586831	+	0.809709i	$0.699625\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−25.2446	−2.02769
$156$	0	0
$157$	7.21110	0.575509	0.287754	−	0.957704i	$-0.407091\pi$
$157$	7.21110	0.575509	0.287754	+	0.957704i	$0.407091\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.86704	0.462388
$162$	0	0
$163$	−11.8167	−0.925552	−0.462776	−	0.886475i	$-0.653147\pi$
$163$	−11.8167	−0.925552	−0.462776	+	0.886475i	$0.653147\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.1962	−0.866391	−0.433195	−	0.901300i	$-0.642614\pi$
$167$	−11.1962	−0.866391	−0.433195	+	0.901300i	$0.642614\pi$
$168$	0	0
$169$	−12.6333	−0.971793
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.31425	0.175949	0.0879745	−	0.996123i	$-0.471961\pi$
$173$	2.31425	0.175949	0.0879745	+	0.996123i	$0.471961\pi$
$174$	0	0
$175$	9.60555	0.726111
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.04532	0.152875	0.0764373	−	0.997074i	$-0.475645\pi$
$179$	2.04532	0.152875	0.0764373	+	0.997074i	$0.475645\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.64344	0.561957
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.86704	−0.424524	−0.212262	−	0.977213i	$-0.568083\pi$
$191$	−5.86704	−0.424524	−0.212262	+	0.977213i	$0.568083\pi$
$192$	0	0
$193$	−15.2111	−1.09492	−0.547460	−	0.836832i	$-0.684405\pi$
$193$	−15.2111	−1.09492	−0.547460	+	0.836832i	$0.684405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−15.2869	−1.08915	−0.544573	−	0.838714i	$-0.683308\pi$
$197$	−15.2869	−1.08915	−0.544573	+	0.838714i	$0.683308\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.04532	−0.143554
$204$	0	0
$205$	22.4222	1.56603
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.86704	0.405832
$210$	0	0
$211$	25.2111	1.73560	0.867802	−	0.496910i	$-0.165532\pi$
$211$	25.2111	1.73560	0.867802	+	0.496910i	$0.165532\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−25.2446	−1.72167
$216$	0	0
$217$	−6.60555	−0.448414
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.23855	0.0833138
$222$	0	0
$223$	−22.2389	−1.48922	−0.744612	−	0.667497i	$-0.767365\pi$
$223$	−22.2389	−1.48922	−0.744612	+	0.667497i	$0.767365\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.09065	−0.271506	−0.135753	−	0.990743i	$-0.543345\pi$
$227$	−4.09065	−0.271506	−0.135753	+	0.990743i	$0.543345\pi$
$228$	0	0
$229$	−3.21110	−0.212196	−0.106098	−	0.994356i	$-0.533836\pi$
$229$	−3.21110	−0.212196	−0.106098	+	0.994356i	$0.533836\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.5588	1.80544	0.902719	−	0.430230i	$-0.141568\pi$
$233$	27.5588	1.80544	0.902719	+	0.430230i	$0.141568\pi$
$234$	0	0
$235$	7.81665	0.509902
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.95769	0.644109	0.322055	−	0.946721i	$-0.395626\pi$
$239$	9.95769	0.644109	0.322055	+	0.946721i	$0.395626\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.82172	0.244161
$246$	0	0
$247$	−0.605551	−0.0385303
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.9756	−1.57645	−0.788224	−	0.615388i	$-0.788999\pi$
$251$	−24.9756	−1.57645	−0.788224	+	0.615388i	$0.788999\pi$
$252$	0	0
$253$	34.4222	2.16411
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.77640	−0.110809	−0.0554043	−	0.998464i	$-0.517645\pi$
$257$	−1.77640	−0.110809	−0.0554043	+	0.998464i	$0.517645\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.5105	−0.833092	−0.416546	−	0.909115i	$-0.636760\pi$
$263$	−13.5105	−0.833092	−0.416546	+	0.909115i	$0.636760\pi$
$264$	0	0
$265$	14.6056	0.897212
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.40489	0.390513	0.195257	−	0.980752i	$-0.437446\pi$
$269$	6.40489	0.390513	0.195257	+	0.980752i	$0.437446\pi$
$270$	0	0
$271$	1.21110	0.0735692	0.0367846	−	0.999323i	$-0.488288\pi$
$271$	1.21110	0.0735692	0.0367846	+	0.999323i	$0.488288\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	56.3562	3.39841
$276$	0	0
$277$	27.0278	1.62394	0.811970	−	0.583699i	$-0.198395\pi$
$277$	27.0278	1.62394	0.811970	+	0.583699i	$0.198395\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.6614	−1.35187	−0.675933	−	0.736963i	$-0.736259\pi$
$281$	−22.6614	−1.35187	−0.675933	+	0.736963i	$0.736259\pi$
$282$	0	0
$283$	−19.6333	−1.16708	−0.583539	−	0.812085i	$-0.698333\pi$
$283$	−19.6333	−1.16708	−0.583539	+	0.812085i	$0.698333\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.86704	0.346321
$288$	0	0
$289$	−12.8167	−0.753921
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.6011	1.02827	0.514135	−	0.857710i	$-0.328113\pi$
$293$	17.6011	1.02827	0.514135	+	0.857710i	$0.328113\pi$
$294$	0	0
$295$	44.8444	2.61094
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.55280	−0.205463
$300$	0	0
$301$	−6.60555	−0.380738
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.64344	0.437662
$306$	0	0
$307$	21.0278	1.20012	0.600059	−	0.799956i	$-0.295144\pi$
$307$	21.0278	1.20012	0.600059	+	0.799956i	$0.295144\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.8277	−1.57797	−0.788983	−	0.614415i	$-0.789392\pi$
$311$	−27.8277	−1.57797	−0.788983	+	0.614415i	$0.789392\pi$
$312$	0	0
$313$	19.2111	1.08588	0.542938	−	0.839773i	$-0.317312\pi$
$313$	19.2111	1.08588	0.542938	+	0.839773i	$0.317312\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.59812	−0.314422	−0.157211	−	0.987565i	$-0.550250\pi$
$317$	−5.59812	−0.314422	−0.157211	+	0.987565i	$0.550250\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−2.04532	−0.113805
$324$	0	0
$325$	−5.81665	−0.322650
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.04532	0.112762
$330$	0	0
$331$	−21.2111	−1.16587	−0.582934	−	0.812520i	$-0.698095\pi$
$331$	−21.2111	−1.16587	−0.582934	+	0.812520i	$0.698095\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−35.2023	−1.92331
$336$	0	0
$337$	34.8444	1.89810	0.949048	−	0.315132i	$-0.102049\pi$
$337$	34.8444	1.89810	0.949048	+	0.315132i	$0.102049\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−38.7551	−2.09870
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.3502	1.73665	0.868324	−	0.495998i	$-0.165197\pi$
$347$	32.3502	1.73665	0.868324	+	0.495998i	$0.165197\pi$
$348$	0	0
$349$	19.2111	1.02835	0.514173	−	0.857686i	$-0.328099\pi$
$349$	19.2111	1.02835	0.514173	+	0.857686i	$0.328099\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.3048	1.61296	0.806482	−	0.591259i	$-0.201369\pi$
$353$	30.3048	1.61296	0.806482	+	0.591259i	$0.201369\pi$
$354$	0	0
$355$	−59.4500	−3.15528
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.31425	0.122141	0.0610707	−	0.998133i	$-0.480548\pi$
$359$	2.31425	0.122141	0.0610707	+	0.998133i	$0.480548\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	27.5588	1.44249
$366$	0	0
$367$	2.78890	0.145579	0.0727896	−	0.997347i	$-0.476810\pi$
$367$	2.78890	0.145579	0.0727896	+	0.997347i	$0.476810\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.82172	0.198414
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.23855	0.0637885
$378$	0	0
$379$	−36.8444	−1.89257	−0.946285	−	0.323333i	$-0.895196\pi$
$379$	−36.8444	−1.89257	−0.946285	+	0.323333i	$0.895196\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.9303	−1.17168	−0.585842	−	0.810425i	$-0.699236\pi$
$383$	−22.9303	−1.17168	−0.585842	+	0.810425i	$0.699236\pi$
$384$	0	0
$385$	22.4222	1.14274
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.9303	1.16261	0.581307	−	0.813684i	$-0.302542\pi$
$389$	22.9303	1.16261	0.581307	+	0.813684i	$0.302542\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.2869	−0.769167
$396$	0	0
$397$	24.4222	1.22572	0.612858	−	0.790193i	$-0.290020\pi$
$397$	24.4222	1.22572	0.612858	+	0.790193i	$0.290020\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.0937	−0.803679	−0.401839	−	0.915710i	$-0.631629\pi$
$401$	−16.0937	−0.803679	−0.401839	+	0.915710i	$0.631629\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	11.7341	0.581637
$408$	0	0
$409$	21.8167	1.07876	0.539382	−	0.842061i	$-0.318658\pi$
$409$	21.8167	1.07876	0.539382	+	0.842061i	$0.318658\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.7341	0.577397
$414$	0	0
$415$	−14.6056	−0.716958
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.50747	0.0736449	0.0368224	−	0.999322i	$-0.488276\pi$
$419$	1.50747	0.0736449	0.0368224	+	0.999322i	$0.488276\pi$
$420$	0	0
$421$	19.2111	0.936292	0.468146	−	0.883651i	$-0.344922\pi$
$421$	19.2111	0.936292	0.468146	+	0.883651i	$0.344922\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−19.6465	−0.952993
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.0663	−1.40007	−0.700037	−	0.714106i	$-0.746833\pi$
$431$	−29.0663	−1.40007	−0.700037	+	0.714106i	$0.746833\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.86704	0.280659
$438$	0	0
$439$	33.0278	1.57633	0.788164	−	0.615465i	$-0.211032\pi$
$439$	33.0278	1.57633	0.788164	+	0.615465i	$0.211032\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.88199	0.421996	0.210998	−	0.977486i	$-0.432329\pi$
$443$	8.88199	0.421996	0.210998	+	0.977486i	$0.432329\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.82172	0.180358	0.0901791	−	0.995926i	$-0.471256\pi$
$449$	3.82172	0.180358	0.0901791	+	0.995926i	$0.471256\pi$
$450$	0	0
$451$	34.4222	1.62088
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.31425	−0.108494
$456$	0	0
$457$	−35.0278	−1.63853	−0.819265	−	0.573416i	$-0.805618\pi$
$457$	−35.0278	−1.63853	−0.819265	+	0.573416i	$0.805618\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	32.6191	1.51922	0.759611	−	0.650377i	$-0.225389\pi$
$461$	32.6191	1.51922	0.759611	+	0.650377i	$0.225389\pi$
$462$	0	0
$463$	−38.4222	−1.78563	−0.892816	−	0.450422i	$-0.851273\pi$
$463$	−38.4222	−1.78563	−0.892816	+	0.450422i	$0.851273\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.5558	0.719837	0.359918	−	0.932984i	$-0.382804\pi$
$467$	15.5558	0.719837	0.359918	+	0.932984i	$0.382804\pi$
$468$	0	0
$469$	−9.21110	−0.425329
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−38.7551	−1.78196
$474$	0	0
$475$	9.60555	0.440733
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.13597	0.280360	0.140180	−	0.990126i	$-0.455232\pi$
$479$	6.13597	0.280360	0.140180	+	0.990126i	$0.455232\pi$
$480$	0	0
$481$	−1.21110	−0.0552215
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.2719	−0.557240
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.88199	0.400839	0.200419	−	0.979710i	$-0.435770\pi$
$491$	8.88199	0.400839	0.200419	+	0.979710i	$0.435770\pi$
$492$	0	0
$493$	4.18335	0.188408
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.5558	−0.697773
$498$	0	0
$499$	18.4222	0.824691	0.412346	−	0.911028i	$-0.364710\pi$
$499$	18.4222	0.824691	0.412346	+	0.911028i	$0.364710\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.0180	−0.669618	−0.334809	−	0.942286i	$-0.608672\pi$
$503$	−15.0180	−0.669618	−0.334809	+	0.942286i	$0.608672\pi$
$504$	0	0
$505$	−23.4500	−1.04351
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.9787	−1.63905	−0.819525	−	0.573043i	$-0.805763\pi$
$509$	−36.9787	−1.63905	−0.819525	+	0.573043i	$0.805763\pi$
$510$	0	0
$511$	7.21110	0.319000
$512$	0	0
$513$	0	0
$514$	0	0
$515$	70.4045	3.10239
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.40489	−0.280604	−0.140302	−	0.990109i	$-0.544807\pi$
$521$	−6.40489	−0.280604	−0.140302	+	0.990109i	$0.544807\pi$
$522$	0	0
$523$	−23.8167	−1.04143	−0.520715	−	0.853731i	$-0.674334\pi$
$523$	−23.8167	−1.04143	−0.520715	+	0.853731i	$0.674334\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.5105	0.588526
$528$	0	0
$529$	11.4222	0.496618
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.55280	−0.153889
$534$	0	0
$535$	−7.81665	−0.337943
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.86704	0.252711
$540$	0	0
$541$	8.78890	0.377864	0.188932	−	0.981990i	$-0.439497\pi$
$541$	8.78890	0.377864	0.188932	+	0.981990i	$0.439497\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.64344	0.327409
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.04532	−0.0871337
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.5588	1.16770	0.583852	−	0.811860i	$-0.301545\pi$
$557$	27.5588	1.16770	0.583852	+	0.811860i	$0.301545\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.1116	1.31120	0.655599	−	0.755109i	$-0.272416\pi$
$563$	31.1116	1.31120	0.655599	+	0.755109i	$0.272416\pi$
$564$	0	0
$565$	−52.6611	−2.21547
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.91237	0.331704	0.165852	−	0.986151i	$-0.446963\pi$
$569$	7.91237	0.331704	0.165852	+	0.986151i	$0.446963\pi$
$570$	0	0
$571$	−17.5778	−0.735608	−0.367804	−	0.929903i	$-0.619890\pi$
$571$	−17.5778	−0.735608	−0.367804	+	0.929903i	$0.619890\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	56.3562	2.35022
$576$	0	0
$577$	−1.63331	−0.0679955	−0.0339977	−	0.999422i	$-0.510824\pi$
$577$	−1.63331	−0.0679955	−0.0339977	+	0.999422i	$0.510824\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.82172	−0.158552
$582$	0	0
$583$	22.4222	0.928633
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−33.1569	−1.36853	−0.684267	−	0.729232i	$-0.739878\pi$
$587$	−33.1569	−1.36853	−0.684267	+	0.729232i	$0.739878\pi$
$588$	0	0
$589$	−6.60555	−0.272177
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.28387	−0.134852	−0.0674262	−	0.997724i	$-0.521479\pi$
$593$	−3.28387	−0.134852	−0.0674262	+	0.997724i	$0.521479\pi$
$594$	0	0
$595$	−7.81665	−0.320452
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.4378	0.998501	0.499251	−	0.866458i	$-0.333609\pi$
$599$	24.4378	0.998501	0.499251	+	0.866458i	$0.333609\pi$
$600$	0	0
$601$	32.7889	1.33749	0.668744	−	0.743493i	$-0.266832\pi$
$601$	32.7889	1.33749	0.668744	+	0.743493i	$0.266832\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	89.5131	3.63923
$606$	0	0
$607$	16.8444	0.683694	0.341847	−	0.939756i	$-0.388948\pi$
$607$	16.8444	0.683694	0.341847	+	0.939756i	$0.388948\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.23855	−0.0501063
$612$	0	0
$613$	−19.3944	−0.783334	−0.391667	−	0.920107i	$-0.628102\pi$
$613$	−19.3944	−0.783334	−0.391667	+	0.920107i	$0.628102\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.55280	−0.143030	−0.0715151	−	0.997440i	$-0.522783\pi$
$617$	−3.55280	−0.143030	−0.0715151	+	0.997440i	$0.522783\pi$
$618$	0	0
$619$	37.2111	1.49564	0.747820	−	0.663901i	$-0.231100\pi$
$619$	37.2111	1.49564	0.747820	+	0.663901i	$0.231100\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.41984	−0.377398
$624$	0	0
$625$	19.2389	0.769554
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.09065	−0.163105
$630$	0	0
$631$	−18.6056	−0.740675	−0.370338	−	0.928897i	$-0.620758\pi$
$631$	−18.6056	−0.740675	−0.370338	+	0.928897i	$0.620758\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	50.4891	2.00360
$636$	0	0
$637$	−0.605551	−0.0239928
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.2266	0.403927	0.201964	−	0.979393i	$-0.435268\pi$
$641$	10.2266	0.403927	0.201964	+	0.979393i	$0.435268\pi$
$642$	0	0
$643$	46.0555	1.81625	0.908126	−	0.418697i	$-0.137513\pi$
$643$	46.0555	1.81625	0.908126	+	0.418697i	$0.137513\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.04532	0.0804099	0.0402050	−	0.999191i	$-0.487199\pi$
$647$	2.04532	0.0804099	0.0402050	+	0.999191i	$0.487199\pi$
$648$	0	0
$649$	68.8444	2.70238
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.8397	−0.737254	−0.368627	−	0.929577i	$-0.620172\pi$
$653$	−18.8397	−0.737254	−0.368627	+	0.929577i	$0.620172\pi$
$654$	0	0
$655$	−43.8167	−1.71206
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.268925	0.0104758	0.00523792	−	0.999986i	$-0.498333\pi$
$659$	0.268925	0.0104758	0.00523792	+	0.999986i	$0.498333\pi$
$660$	0	0
$661$	−41.8167	−1.62648	−0.813240	−	0.581929i	$-0.802298\pi$
$661$	−41.8167	−1.62648	−0.813240	+	0.581929i	$0.802298\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.82172	0.148200
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.7341	0.452989
$672$	0	0
$673$	5.63331	0.217148	0.108574	−	0.994088i	$-0.465372\pi$
$673$	5.63331	0.217148	0.108574	+	0.994088i	$0.465372\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	36.9787	1.42121	0.710603	−	0.703593i	$-0.248422\pi$
$677$	36.9787	1.42121	0.710603	+	0.703593i	$0.248422\pi$
$678$	0	0
$679$	−3.21110	−0.123231
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.04532	0.0782621	0.0391311	−	0.999234i	$-0.487541\pi$
$683$	2.04532	0.0782621	0.0391311	+	0.999234i	$0.487541\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.31425	−0.0881658
$690$	0	0
$691$	−34.7889	−1.32343	−0.661716	−	0.749755i	$-0.730172\pi$
$691$	−34.7889	−1.32343	−0.661716	+	0.749755i	$0.730172\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.6584	0.404296
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8.18129	0.309003	0.154502	−	0.987993i	$-0.450623\pi$
$701$	8.18129	0.309003	0.154502	+	0.987993i	$0.450623\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.13597	−0.230767
$708$	0	0
$709$	−33.4500	−1.25624	−0.628120	−	0.778117i	$-0.716175\pi$
$709$	−33.4500	−1.25624	−0.628120	+	0.778117i	$0.716175\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−38.7551	−1.45139
$714$	0	0
$715$	−13.5778	−0.507781
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−39.0240	−1.45535	−0.727675	−	0.685923i	$-0.759399\pi$
$719$	−39.0240	−1.45535	−0.727675	+	0.685923i	$0.759399\pi$
$720$	0	0
$721$	18.4222	0.686079
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−19.6465	−0.729651
$726$	0	0
$727$	−14.4222	−0.534890	−0.267445	−	0.963573i	$-0.586179\pi$
$727$	−14.4222	−0.534890	−0.267445	+	0.963573i	$0.586179\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13.5105	0.499703
$732$	0	0
$733$	−28.7889	−1.06334	−0.531671	−	0.846951i	$-0.678436\pi$
$733$	−28.7889	−1.06334	−0.531671	+	0.846951i	$0.678436\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−54.0419	−1.99066
$738$	0	0
$739$	−46.2389	−1.70092	−0.850462	−	0.526037i	$-0.823678\pi$
$739$	−46.2389	−1.70092	−0.850462	+	0.526037i	$0.823678\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.68876	0.355446	0.177723	−	0.984081i	$-0.443127\pi$
$743$	9.68876	0.355446	0.177723	+	0.984081i	$0.443127\pi$
$744$	0	0
$745$	29.2111	1.07021
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.04532	−0.0747345
$750$	0	0
$751$	−48.8444	−1.78236	−0.891179	−	0.453652i	$-0.850121\pi$
$751$	−48.8444	−1.78236	−0.891179	+	0.453652i	$0.850121\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−55.1176	−2.00594
$756$	0	0
$757$	43.2111	1.57053	0.785267	−	0.619157i	$-0.212526\pi$
$757$	43.2111	1.57053	0.785267	+	0.619157i	$0.212526\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.3048	−1.09855	−0.549275	−	0.835642i	$-0.685096\pi$
$761$	−30.3048	−1.09855	−0.549275	+	0.835642i	$0.685096\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7.10559	−0.256568
$768$	0	0
$769$	−15.2111	−0.548526	−0.274263	−	0.961655i	$-0.588434\pi$
$769$	−15.2111	−0.548526	−0.274263	+	0.961655i	$0.588434\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−43.5464	−1.56626	−0.783128	−	0.621861i	$-0.786377\pi$
$773$	−43.5464	−1.56626	−0.783128	+	0.621861i	$0.786377\pi$
$774$	0	0
$775$	−63.4500	−2.27919
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.86704	0.210209
$780$	0	0
$781$	−91.2666	−3.26578
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.5588	0.983617
$786$	0	0
$787$	−42.6056	−1.51872	−0.759362	−	0.650668i	$-0.774489\pi$
$787$	−42.6056	−1.51872	−0.759362	+	0.650668i	$0.774489\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.7794	−0.489939
$792$	0	0
$793$	−1.21110	−0.0430075
$794$	0	0
$795$	0	0
$796$	0	0
$797$	32.8880	1.16495	0.582477	−	0.812847i	$-0.302084\pi$
$797$	32.8880	1.16495	0.582477	+	0.812847i	$0.302084\pi$
$798$	0	0
$799$	−4.18335	−0.147996
$800$	0	0
$801$	0	0
$802$	0	0
$803$	42.3079	1.49301
$804$	0	0
$805$	22.4222	0.790279
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.3626	−0.575278	−0.287639	−	0.957739i	$-0.592870\pi$
$809$	−16.3626	−0.575278	−0.287639	+	0.957739i	$0.592870\pi$
$810$	0	0
$811$	−17.5778	−0.617240	−0.308620	−	0.951185i	$-0.599867\pi$
$811$	−17.5778	−0.617240	−0.308620	+	0.951185i	$0.599867\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−45.1600	−1.58188
$816$	0	0
$817$	−6.60555	−0.231099
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−49.9513	−1.74331	−0.871656	−	0.490118i	$-0.836954\pi$
$821$	−49.9513	−1.74331	−0.871656	+	0.490118i	$0.836954\pi$
$822$	0	0
$823$	−8.18335	−0.285254	−0.142627	−	0.989777i	$-0.545555\pi$
$823$	−8.18335	−0.285254	−0.142627	+	0.989777i	$0.545555\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.28387	0.114191	0.0570957	−	0.998369i	$-0.481816\pi$
$827$	3.28387	0.114191	0.0570957	+	0.998369i	$0.481816\pi$
$828$	0	0
$829$	29.6333	1.02921	0.514604	−	0.857428i	$-0.327939\pi$
$829$	29.6333	1.02921	0.514604	+	0.857428i	$0.327939\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.04532	−0.0708662
$834$	0	0
$835$	−42.7889	−1.48077
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.6794	−1.30084	−0.650418	−	0.759576i	$-0.725406\pi$
$839$	−37.6794	−1.30084	−0.650418	+	0.759576i	$0.725406\pi$
$840$	0	0
$841$	−24.8167	−0.855747
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−48.2810	−1.66092
$846$	0	0
$847$	23.4222	0.804796
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11.7341	0.402239
$852$	0	0
$853$	53.6333	1.83637	0.918185	−	0.396152i	$-0.129655\pi$
$853$	53.6333	1.83637	0.918185	+	0.396152i	$0.129655\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.86704	−0.200414	−0.100207	−	0.994967i	$-0.531951\pi$
$857$	−5.86704	−0.200414	−0.100207	+	0.994967i	$0.531951\pi$
$858$	0	0
$859$	1.21110	0.0413223	0.0206611	−	0.999787i	$-0.493423\pi$
$859$	1.21110	0.0413223	0.0206611	+	0.999787i	$0.493423\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.68876	−0.329809	−0.164905	−	0.986310i	$-0.552732\pi$
$863$	−9.68876	−0.329809	−0.164905	+	0.986310i	$0.552732\pi$
$864$	0	0
$865$	8.84441	0.300719
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.4682	−0.796103
$870$	0	0
$871$	5.57779	0.188996
$872$	0	0
$873$	0	0
$874$	0	0
$875$	17.6011	0.595027
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0.806775	0.0271809	0.0135905	−	0.999908i	$-0.495674\pi$
$881$	0.806775	0.0271809	0.0135905	+	0.999908i	$0.495674\pi$
$882$	0	0
$883$	28.8444	0.970692	0.485346	−	0.874322i	$-0.338694\pi$
$883$	28.8444	0.970692	0.485346	+	0.874322i	$0.338694\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.4532	0.686752	0.343376	−	0.939198i	$-0.388429\pi$
$887$	20.4532	0.686752	0.343376	+	0.939198i	$0.388429\pi$
$888$	0	0
$889$	13.2111	0.443086
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.04532	0.0684441
$894$	0	0
$895$	7.81665	0.261282
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.5105	0.450600
$900$	0	0
$901$	−7.81665	−0.260410
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.64344	0.254077
$906$	0	0
$907$	51.2666	1.70228	0.851140	−	0.524939i	$-0.175912\pi$
$907$	51.2666	1.70228	0.851140	+	0.524939i	$0.175912\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−11.4652	−0.379858	−0.189929	−	0.981798i	$-0.560826\pi$
$911$	−11.4652	−0.379858	−0.189929	+	0.981798i	$0.560826\pi$
$912$	0	0
$913$	−22.4222	−0.742067
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.4652	−0.378613
$918$	0	0
$919$	18.4222	0.607692	0.303846	−	0.952721i	$-0.401729\pi$
$919$	18.4222	0.607692	0.303846	+	0.952721i	$0.401729\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9.41984	0.310058
$924$	0	0
$925$	19.2111	0.631657
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−29.0663	−0.953634	−0.476817	−	0.879003i	$-0.658210\pi$
$929$	−29.0663	−0.953634	−0.476817	+	0.879003i	$0.658210\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−45.8607	−1.49980
$936$	0	0
$937$	15.5778	0.508904	0.254452	−	0.967085i	$-0.418105\pi$
$937$	15.5778	0.508904	0.254452	+	0.967085i	$0.418105\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−55.8183	−1.81963	−0.909813	−	0.415019i	$-0.863775\pi$
$941$	−55.8183	−1.81963	−0.909813	+	0.415019i	$0.863775\pi$
$942$	0	0
$943$	34.4222	1.12094
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.6011	−0.571960	−0.285980	−	0.958236i	$-0.592319\pi$
$947$	−17.6011	−0.571960	−0.285980	+	0.958236i	$0.592319\pi$
$948$	0	0
$949$	−4.36669	−0.141749
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−43.1146	−1.39662	−0.698310	−	0.715796i	$-0.746064\pi$
$953$	−43.1146	−1.39662	−0.698310	+	0.715796i	$0.746064\pi$
$954$	0	0
$955$	−22.4222	−0.725566
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	12.6333	0.407526
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−58.1326	−1.87135
$966$	0	0
$967$	−20.6611	−0.664415	−0.332208	−	0.943206i	$-0.607793\pi$
$967$	−20.6611	−0.664415	−0.332208	+	0.943206i	$0.607793\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.9453	0.832623	0.416312	−	0.909222i	$-0.363323\pi$
$971$	25.9453	0.832623	0.416312	+	0.909222i	$0.363323\pi$
$972$	0	0
$973$	2.78890	0.0894079
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53.7730	−1.72035	−0.860176	−	0.509998i	$-0.829646\pi$
$977$	−53.7730	−1.72035	−0.860176	+	0.509998i	$0.829646\pi$
$978$	0	0
$979$	−55.2666	−1.76633
$980$	0	0
$981$	0	0
$982$	0	0
$983$	51.5648	1.64466	0.822332	−	0.569009i	$-0.192673\pi$
$983$	51.5648	1.64466	0.822332	+	0.569009i	$0.192673\pi$
$984$	0	0
$985$	−58.4222	−1.86149
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−38.7551	−1.23234
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30.5738	0.969254
$996$	0	0
$997$	−37.6333	−1.19186	−0.595929	−	0.803037i	$-0.703216\pi$
$997$	−37.6333	−1.19186	−0.595929	+	0.803037i	$0.703216\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.s.1.4	yes	4
3.2	odd	2	inner	4788.2.a.s.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.s.1.1	✓	4	3.2	odd	2	inner
4788.2.a.s.1.4	yes	4	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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+3.82172 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+3.82172 q^{5} +1.00000 q^{7} +5.86704 q^{11} -0.605551 q^{13} -2.04532 q^{17} +1.00000 q^{19} +5.86704 q^{23} +9.60555 q^{25} -2.04532 q^{29} -6.60555 q^{31} +3.82172 q^{35} +2.00000 q^{37} +5.86704 q^{41} -6.60555 q^{43} +2.04532 q^{47} +1.00000 q^{49} +3.82172 q^{53} +22.4222 q^{55} +11.7341 q^{59} +2.00000 q^{61} -2.31425 q^{65} -9.21110 q^{67} -15.5558 q^{71} +7.21110 q^{73} +5.86704 q^{77} -4.00000 q^{79} -3.82172 q^{83} -7.81665 q^{85} -9.41984 q^{89} -0.605551 q^{91} +3.82172 q^{95} -3.21110 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 12 q^{13} + 4 q^{19} + 24 q^{25} - 12 q^{31} + 8 q^{37} - 12 q^{43} + 4 q^{49} + 32 q^{55} + 8 q^{61} - 8 q^{67} - 16 q^{79} + 12 q^{85} + 12 q^{91} + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 12 * q^13 + 4 * q^19 + 24 * q^25 - 12 * q^31 + 8 * q^37 - 12 * q^43 + 4 * q^49 + 32 * q^55 + 8 * q^61 - 8 * q^67 - 16 * q^79 + 12 * q^85 + 12 * q^91 + 16 * q^97

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