LMFDB - Embedded newform 4788.2.a.q.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4788,2,Mod(1,4788)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,0,0,0,0,-4,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.2323724878$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.28825$ 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)

$f(q)$	$=$	$q+2.28825 q^{5} -1.00000 q^{7} -2.82843 q^{11} -3.23607 q^{13} +0.540182 q^{17} +1.00000 q^{19} +2.82843 q^{23} +0.236068 q^{25} -8.61280 q^{29} +7.70820 q^{31} -2.28825 q^{35} +4.47214 q^{37} -8.48528 q^{41} -5.23607 q^{43} -4.03631 q^{47} +1.00000 q^{49} +11.4412 q^{53} -6.47214 q^{55} +2.16073 q^{59} -3.52786 q^{61} -7.40492 q^{65} -15.4164 q^{67} +12.5216 q^{71} -10.9443 q^{73} +2.82843 q^{77} -10.4721 q^{79} -7.94510 q^{83} +1.23607 q^{85} -8.48528 q^{89} +3.23607 q^{91} +2.28825 q^{95} -12.4721 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 4 q^{13} + 4 q^{19} - 8 q^{25} + 4 q^{31} - 12 q^{43} + 4 q^{49} - 8 q^{55} - 32 q^{61} - 8 q^{67} - 8 q^{73} - 24 q^{79} - 4 q^{85} + 4 q^{91} - 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^13 + 4 * q^19 - 8 * q^25 + 4 * q^31 - 12 * q^43 + 4 * q^49 - 8 * q^55 - 32 * q^61 - 8 * q^67 - 8 * q^73 - 24 * q^79 - 4 * q^85 + 4 * q^91 - 32 * 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
$5$	2.28825	1.02333	0.511667	−	0.859184i	$-0.329028\pi$
$5$	2.28825	1.02333	0.511667	+	0.859184i	$0.329028\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.540182	0.131013	0.0655066	−	0.997852i	$-0.479134\pi$
$17$	0.540182	0.131013	0.0655066	+	0.997852i	$0.479134\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	0.236068	0.0472136
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.61280	−1.59936	−0.799678	−	0.600428i	$-0.794997\pi$
$29$	−8.61280	−1.59936	−0.799678	+	0.600428i	$0.794997\pi$
$30$	0	0
$31$	7.70820	1.38443	0.692217	−	0.721689i	$-0.256634\pi$
$31$	7.70820	1.38443	0.692217	+	0.721689i	$0.256634\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.28825	−0.386784
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.48528	−1.32518	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528	−1.32518	−0.662589	+	0.748983i	$0.730542\pi$
$42$	0	0
$43$	−5.23607	−0.798493	−0.399246	−	0.916844i	$-0.630728\pi$
$43$	−5.23607	−0.798493	−0.399246	+	0.916844i	$0.630728\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.03631	−0.588756	−0.294378	−	0.955689i	$-0.595113\pi$
$47$	−4.03631	−0.588756	−0.294378	+	0.955689i	$0.595113\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.4412	1.57157	0.785787	−	0.618497i	$-0.212258\pi$
$53$	11.4412	1.57157	0.785787	+	0.618497i	$0.212258\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.16073	0.281303	0.140651	−	0.990059i	$-0.455080\pi$
$59$	2.16073	0.281303	0.140651	+	0.990059i	$0.455080\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.40492	−0.918467
$66$	0	0
$67$	−15.4164	−1.88341	−0.941707	−	0.336434i	$-0.890779\pi$
$67$	−15.4164	−1.88341	−0.941707	+	0.336434i	$0.890779\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.5216	1.48604	0.743020	−	0.669270i	$-0.233393\pi$
$71$	12.5216	1.48604	0.743020	+	0.669270i	$0.233393\pi$
$72$	0	0
$73$	−10.9443	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$73$	−10.9443	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	−10.4721	−1.17821	−0.589104	−	0.808057i	$-0.700519\pi$
$79$	−10.4721	−1.17821	−0.589104	+	0.808057i	$0.700519\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.94510	−0.872088	−0.436044	−	0.899925i	$-0.643621\pi$
$83$	−7.94510	−0.872088	−0.436044	+	0.899925i	$0.643621\pi$
$84$	0	0
$85$	1.23607	0.134070
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.48528	−0.899438	−0.449719	−	0.893170i	$-0.648476\pi$
$89$	−8.48528	−0.899438	−0.449719	+	0.893170i	$0.648476\pi$
$90$	0	0
$91$	3.23607	0.339232
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.28825	0.234769
$96$	0	0
$97$	−12.4721	−1.26635	−0.633177	−	0.774007i	$-0.718249\pi$
$97$	−12.4721	−1.26635	−0.633177	+	0.774007i	$0.718249\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.70091	−0.268750	−0.134375	−	0.990931i	$-0.542903\pi$
$101$	−2.70091	−0.268750	−0.134375	+	0.990931i	$0.542903\pi$
$102$	0	0
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.53244	0.728188	0.364094	−	0.931362i	$-0.381379\pi$
$107$	7.53244	0.728188	0.364094	+	0.931362i	$0.381379\pi$
$108$	0	0
$109$	−1.05573	−0.101120	−0.0505602	−	0.998721i	$-0.516101\pi$
$109$	−1.05573	−0.101120	−0.0505602	+	0.998721i	$0.516101\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.2697	−1.34238	−0.671188	−	0.741287i	$-0.734216\pi$
$113$	−14.2697	−1.34238	−0.671188	+	0.741287i	$0.734216\pi$
$114$	0	0
$115$	6.47214	0.603530
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.540182	−0.0495184
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.9010	−0.975019
$126$	0	0
$127$	−19.4164	−1.72293	−0.861464	−	0.507819i	$-0.830452\pi$
$127$	−19.4164	−1.72293	−0.861464	+	0.507819i	$0.830452\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.5216	−1.09402	−0.547008	−	0.837127i	$-0.684233\pi$
$131$	−12.5216	−1.09402	−0.547008	+	0.837127i	$0.684233\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.8098	1.26529	0.632645	−	0.774442i	$-0.281969\pi$
$137$	14.8098	1.26529	0.632645	+	0.774442i	$0.281969\pi$
$138$	0	0
$139$	12.9443	1.09792	0.548959	−	0.835849i	$-0.315024\pi$
$139$	12.9443	1.09792	0.548959	+	0.835849i	$0.315024\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.15298	0.765411
$144$	0	0
$145$	−19.7082	−1.63668
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.7295	−1.12476	−0.562381	−	0.826878i	$-0.690115\pi$
$149$	−13.7295	−1.12476	−0.562381	+	0.826878i	$0.690115\pi$
$150$	0	0
$151$	−10.4721	−0.852210	−0.426105	−	0.904674i	$-0.640115\pi$
$151$	−10.4721	−0.852210	−0.426105	+	0.904674i	$0.640115\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.6383	1.41674
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.82843	−0.222911
$162$	0	0
$163$	15.1246	1.18465	0.592326	−	0.805699i	$-0.298210\pi$
$163$	15.1246	1.18465	0.592326	+	0.805699i	$0.298210\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.2333	0.791880	0.395940	−	0.918276i	$-0.370419\pi$
$167$	10.2333	0.791880	0.395940	+	0.918276i	$0.370419\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.16383	−0.316570	−0.158285	−	0.987393i	$-0.550596\pi$
$173$	−4.16383	−0.316570	−0.158285	+	0.987393i	$0.550596\pi$
$174$	0	0
$175$	−0.236068	−0.0178451
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.7735	0.805251	0.402626	−	0.915365i	$-0.368098\pi$
$179$	10.7735	0.805251	0.402626	+	0.915365i	$0.368098\pi$
$180$	0	0
$181$	2.94427	0.218846	0.109423	−	0.993995i	$-0.465100\pi$
$181$	2.94427	0.218846	0.109423	+	0.993995i	$0.465100\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.2333	0.752371
$186$	0	0
$187$	−1.52786	−0.111728
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.3168	0.963571	0.481785	−	0.876289i	$-0.339988\pi$
$191$	13.3168	0.963571	0.481785	+	0.876289i	$0.339988\pi$
$192$	0	0
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.4667	1.45819	0.729096	−	0.684412i	$-0.239941\pi$
$197$	20.4667	1.45819	0.729096	+	0.684412i	$0.239941\pi$
$198$	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
$203$	8.61280	0.604500
$204$	0	0
$205$	−19.4164	−1.35610
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.82843	−0.195646
$210$	0	0
$211$	10.4721	0.720932	0.360466	−	0.932772i	$-0.382618\pi$
$211$	10.4721	0.720932	0.360466	+	0.932772i	$0.382618\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.9814	−0.817125
$216$	0	0
$217$	−7.70820	−0.523267
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.74806	−0.117588
$222$	0	0
$223$	5.23607	0.350633	0.175317	−	0.984512i	$-0.443905\pi$
$223$	5.23607	0.350633	0.175317	+	0.984512i	$0.443905\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.5687	0.767845	0.383922	−	0.923365i	$-0.374573\pi$
$227$	11.5687	0.767845	0.383922	+	0.923365i	$0.374573\pi$
$228$	0	0
$229$	−26.3607	−1.74196	−0.870981	−	0.491316i	$-0.836516\pi$
$229$	−26.3607	−1.74196	−0.870981	+	0.491316i	$0.836516\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.3941	0.811963	0.405981	−	0.913881i	$-0.366930\pi$
$233$	12.3941	0.811963	0.405981	+	0.913881i	$0.366930\pi$
$234$	0	0
$235$	−9.23607	−0.602495
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.412662	0.0266929	0.0133464	−	0.999911i	$-0.495752\pi$
$239$	0.412662	0.0266929	0.0133464	+	0.999911i	$0.495752\pi$
$240$	0	0
$241$	11.8885	0.765808	0.382904	−	0.923788i	$-0.374924\pi$
$241$	11.8885	0.765808	0.382904	+	0.923788i	$0.374924\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.28825	0.146191
$246$	0	0
$247$	−3.23607	−0.205906
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.0795	1.83548	0.917741	−	0.397180i	$-0.130011\pi$
$251$	29.0795	1.83548	0.917741	+	0.397180i	$0.130011\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.7186	−1.16764	−0.583818	−	0.811885i	$-0.698442\pi$
$257$	−18.7186	−1.16764	−0.583818	+	0.811885i	$0.698442\pi$
$258$	0	0
$259$	−4.47214	−0.277885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.3755	−1.50306	−0.751528	−	0.659701i	$-0.770683\pi$
$263$	−24.3755	−1.50306	−0.751528	+	0.659701i	$0.770683\pi$
$264$	0	0
$265$	26.1803	1.60825
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.0618	0.796390	0.398195	−	0.917301i	$-0.369637\pi$
$269$	13.0618	0.796390	0.398195	+	0.917301i	$0.369637\pi$
$270$	0	0
$271$	−26.8328	−1.62998	−0.814989	−	0.579477i	$-0.803257\pi$
$271$	−26.8328	−1.62998	−0.814989	+	0.579477i	$0.803257\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.667701	−0.0402639
$276$	0	0
$277$	−24.7639	−1.48792	−0.743960	−	0.668224i	$-0.767055\pi$
$277$	−24.7639	−1.48792	−0.743960	+	0.668224i	$0.767055\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.1058	−0.602863	−0.301432	−	0.953488i	$-0.597464\pi$
$281$	−10.1058	−0.602863	−0.301432	+	0.953488i	$0.597464\pi$
$282$	0	0
$283$	6.47214	0.384729	0.192364	−	0.981324i	$-0.438384\pi$
$283$	6.47214	0.384729	0.192364	+	0.981324i	$0.438384\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.48528	0.500870
$288$	0	0
$289$	−16.7082	−0.982836
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.4698	1.31270	0.656350	−	0.754457i	$-0.272100\pi$
$293$	22.4698	1.31270	0.656350	+	0.754457i	$0.272100\pi$
$294$	0	0
$295$	4.94427	0.287867
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.15298	−0.529331
$300$	0	0
$301$	5.23607	0.301802
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.07262	−0.462237
$306$	0	0
$307$	−3.70820	−0.211638	−0.105819	−	0.994385i	$-0.533746\pi$
$307$	−3.70820	−0.211638	−0.105819	+	0.994385i	$0.533746\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.5216	−0.710034	−0.355017	−	0.934860i	$-0.615525\pi$
$311$	−12.5216	−0.710034	−0.355017	+	0.934860i	$0.615525\pi$
$312$	0	0
$313$	15.8885	0.898074	0.449037	−	0.893513i	$-0.351767\pi$
$313$	15.8885	0.898074	0.449037	+	0.893513i	$0.351767\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.8245	1.61894	0.809471	−	0.587159i	$-0.199754\pi$
$317$	28.8245	1.61894	0.809471	+	0.587159i	$0.199754\pi$
$318$	0	0
$319$	24.3607	1.36394
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.540182	0.0300565
$324$	0	0
$325$	−0.763932	−0.0423753
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.03631	0.222529
$330$	0	0
$331$	−21.8885	−1.20310	−0.601552	−	0.798834i	$-0.705451\pi$
$331$	−21.8885	−1.20310	−0.601552	+	0.798834i	$0.705451\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−35.2765	−1.92736
$336$	0	0
$337$	−6.94427	−0.378279	−0.189139	−	0.981950i	$-0.560570\pi$
$337$	−6.94427	−0.378279	−0.189139	+	0.981950i	$0.560570\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−21.8021	−1.18065
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.8933	−0.960563	−0.480281	−	0.877114i	$-0.659465\pi$
$347$	−17.8933	−0.960563	−0.480281	+	0.877114i	$0.659465\pi$
$348$	0	0
$349$	−17.4164	−0.932279	−0.466139	−	0.884711i	$-0.654356\pi$
$349$	−17.4164	−0.932279	−0.466139	+	0.884711i	$0.654356\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.78437	−0.307871	−0.153936	−	0.988081i	$-0.549195\pi$
$353$	−5.78437	−0.307871	−0.153936	+	0.988081i	$0.549195\pi$
$354$	0	0
$355$	28.6525	1.52072
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.00310	0.105720	0.0528599	−	0.998602i	$-0.483166\pi$
$359$	2.00310	0.105720	0.0528599	+	0.998602i	$0.483166\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−25.0432	−1.31082
$366$	0	0
$367$	−23.4164	−1.22233	−0.611163	−	0.791505i	$-0.709298\pi$
$367$	−23.4164	−1.22233	−0.611163	+	0.791505i	$0.709298\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.4412	−0.593999
$372$	0	0
$373$	−5.05573	−0.261776	−0.130888	−	0.991397i	$-0.541783\pi$
$373$	−5.05573	−0.261776	−0.130888	+	0.991397i	$0.541783\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	27.8716	1.43546
$378$	0	0
$379$	27.4164	1.40829	0.704143	−	0.710058i	$-0.251331\pi$
$379$	27.4164	1.40829	0.704143	+	0.710058i	$0.251331\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	28.5393	1.45829	0.729145	−	0.684359i	$-0.239918\pi$
$383$	28.5393	1.45829	0.729145	+	0.684359i	$0.239918\pi$
$384$	0	0
$385$	6.47214	0.329851
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.3786	1.33745	0.668724	−	0.743511i	$-0.266841\pi$
$389$	26.3786	1.33745	0.668724	+	0.743511i	$0.266841\pi$
$390$	0	0
$391$	1.52786	0.0772674
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−23.9628	−1.20570
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.62365	0.180956	0.0904782	−	0.995898i	$-0.471160\pi$
$401$	3.62365	0.180956	0.0904782	+	0.995898i	$0.471160\pi$
$402$	0	0
$403$	−24.9443	−1.24256
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.6491	−0.626993
$408$	0	0
$409$	−25.1246	−1.24233	−0.621166	−	0.783679i	$-0.713341\pi$
$409$	−25.1246	−1.24233	−0.621166	+	0.783679i	$0.713341\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.16073	−0.106322
$414$	0	0
$415$	−18.1803	−0.892438
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.0717	−1.76222	−0.881110	−	0.472911i	$-0.843203\pi$
$419$	−36.0717	−1.76222	−0.881110	+	0.472911i	$0.843203\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.127520	0.00618561
$426$	0	0
$427$	3.52786	0.170725
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.0286	0.531227	0.265614	−	0.964080i	$-0.414425\pi$
$431$	11.0286	0.531227	0.265614	+	0.964080i	$0.414425\pi$
$432$	0	0
$433$	−11.8885	−0.571327	−0.285663	−	0.958330i	$-0.592214\pi$
$433$	−11.8885	−0.571327	−0.285663	+	0.958330i	$0.592214\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.82843	0.135302
$438$	0	0
$439$	15.1246	0.721858	0.360929	−	0.932593i	$-0.382460\pi$
$439$	15.1246	0.721858	0.360929	+	0.932593i	$0.382460\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.08347	−0.146500	−0.0732500	−	0.997314i	$-0.523337\pi$
$443$	−3.08347	−0.146500	−0.0732500	+	0.997314i	$0.523337\pi$
$444$	0	0
$445$	−19.4164	−0.920426
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−40.8059	−1.92575	−0.962874	−	0.269952i	$-0.912992\pi$
$449$	−40.8059	−1.92575	−0.962874	+	0.269952i	$0.912992\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.40492	0.347148
$456$	0	0
$457$	31.5967	1.47803	0.739017	−	0.673687i	$-0.235290\pi$
$457$	31.5967	1.47803	0.739017	+	0.673687i	$0.235290\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.8307	−1.52908	−0.764538	−	0.644579i	$-0.777033\pi$
$461$	−32.8307	−1.52908	−0.764538	+	0.644579i	$0.777033\pi$
$462$	0	0
$463$	−21.8885	−1.01725	−0.508623	−	0.860989i	$-0.669845\pi$
$463$	−21.8885	−1.01725	−0.508623	+	0.860989i	$0.669845\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.3392	−0.941185	−0.470592	−	0.882351i	$-0.655960\pi$
$467$	−20.3392	−0.941185	−0.470592	+	0.882351i	$0.655960\pi$
$468$	0	0
$469$	15.4164	0.711864
$470$	0	0
$471$	0	0
$472$	0	0
$473$	14.8098	0.680957
$474$	0	0
$475$	0.236068	0.0108315
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.4953	1.43906	0.719528	−	0.694464i	$-0.244358\pi$
$479$	31.4953	1.43906	0.719528	+	0.694464i	$0.244358\pi$
$480$	0	0
$481$	−14.4721	−0.659873
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−28.5393	−1.29590
$486$	0	0
$487$	−2.11146	−0.0956792	−0.0478396	−	0.998855i	$-0.515234\pi$
$487$	−2.11146	−0.0956792	−0.0478396	+	0.998855i	$0.515234\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.5362	−1.19756	−0.598781	−	0.800913i	$-0.704348\pi$
$491$	−26.5362	−1.19756	−0.598781	+	0.800913i	$0.704348\pi$
$492$	0	0
$493$	−4.65248	−0.209537
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.5216	−0.561670
$498$	0	0
$499$	17.8885	0.800801	0.400401	−	0.916340i	$-0.368871\pi$
$499$	17.8885	0.800801	0.400401	+	0.916340i	$0.368871\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6.60970	0.294712	0.147356	−	0.989084i	$-0.452924\pi$
$503$	6.60970	0.294712	0.147356	+	0.989084i	$0.452924\pi$
$504$	0	0
$505$	−6.18034	−0.275022
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.1853	1.73686	0.868429	−	0.495813i	$-0.165130\pi$
$509$	39.1853	1.73686	0.868429	+	0.495813i	$0.165130\pi$
$510$	0	0
$511$	10.9443	0.484146
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.3137	0.498542
$516$	0	0
$517$	11.4164	0.502093
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.90879	0.171247	0.0856236	−	0.996328i	$-0.472712\pi$
$521$	3.90879	0.171247	0.0856236	+	0.996328i	$0.472712\pi$
$522$	0	0
$523$	6.18034	0.270247	0.135124	−	0.990829i	$-0.456857\pi$
$523$	6.18034	0.270247	0.135124	+	0.990829i	$0.456857\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.16383	0.181379
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	27.4589	1.18938
$534$	0	0
$535$	17.2361	0.745180
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	24.4721	1.05214	0.526070	−	0.850441i	$-0.323665\pi$
$541$	24.4721	1.05214	0.526070	+	0.850441i	$0.323665\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.41577	−0.103480
$546$	0	0
$547$	43.4164	1.85635	0.928176	−	0.372142i	$-0.121377\pi$
$547$	43.4164	1.85635	0.928176	+	0.372142i	$0.121377\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.61280	−0.366918
$552$	0	0
$553$	10.4721	0.445321
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.41577	0.102359	0.0511796	−	0.998689i	$-0.483702\pi$
$557$	2.41577	0.102359	0.0511796	+	0.998689i	$0.483702\pi$
$558$	0	0
$559$	16.9443	0.716666
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.40802	−0.396501	−0.198250	−	0.980151i	$-0.563526\pi$
$563$	−9.40802	−0.396501	−0.198250	+	0.980151i	$0.563526\pi$
$564$	0	0
$565$	−32.6525	−1.37370
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.5216	0.524932	0.262466	−	0.964941i	$-0.415464\pi$
$569$	12.5216	0.524932	0.262466	+	0.964941i	$0.415464\pi$
$570$	0	0
$571$	2.11146	0.0883617	0.0441808	−	0.999024i	$-0.485932\pi$
$571$	2.11146	0.0883617	0.0441808	+	0.999024i	$0.485932\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.667701	0.0278451
$576$	0	0
$577$	28.8328	1.20033	0.600163	−	0.799878i	$-0.295102\pi$
$577$	28.8328	1.20033	0.600163	+	0.799878i	$0.295102\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.94510	0.329618
$582$	0	0
$583$	−32.3607	−1.34024
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.62054	0.0668870	0.0334435	−	0.999441i	$-0.489353\pi$
$587$	1.62054	0.0668870	0.0334435	+	0.999441i	$0.489353\pi$
$588$	0	0
$589$	7.70820	0.317611
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.60970	0.271428	0.135714	−	0.990748i	$-0.456667\pi$
$593$	6.60970	0.271428	0.135714	+	0.990748i	$0.456667\pi$
$594$	0	0
$595$	−1.23607	−0.0506738
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.9188	1.09987	0.549935	−	0.835207i	$-0.314652\pi$
$599$	26.9188	1.09987	0.549935	+	0.835207i	$0.314652\pi$
$600$	0	0
$601$	2.58359	0.105387	0.0526935	−	0.998611i	$-0.483219\pi$
$601$	2.58359	0.105387	0.0526935	+	0.998611i	$0.483219\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.86474	−0.279091
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	13.0618	0.528423
$612$	0	0
$613$	−7.81966	−0.315833	−0.157917	−	0.987452i	$-0.550478\pi$
$613$	−7.81966	−0.315833	−0.157917	+	0.987452i	$0.550478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.1375	−0.931480	−0.465740	−	0.884921i	$-0.654212\pi$
$617$	−23.1375	−0.931480	−0.465740	+	0.884921i	$0.654212\pi$
$618$	0	0
$619$	−35.7771	−1.43800	−0.719001	−	0.695009i	$-0.755400\pi$
$619$	−35.7771	−1.43800	−0.719001	+	0.695009i	$0.755400\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.48528	0.339956
$624$	0	0
$625$	−26.1246	−1.04498
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.41577	0.0963229
$630$	0	0
$631$	37.5967	1.49670	0.748351	−	0.663302i	$-0.230846\pi$
$631$	37.5967	1.49670	0.748351	+	0.663302i	$0.230846\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−44.4295	−1.76313
$636$	0	0
$637$	−3.23607	−0.128218
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−14.5247	−0.573691	−0.286845	−	0.957977i	$-0.592607\pi$
$641$	−14.5247	−0.573691	−0.286845	+	0.957977i	$0.592607\pi$
$642$	0	0
$643$	46.4721	1.83268	0.916341	−	0.400399i	$-0.131128\pi$
$643$	46.4721	1.83268	0.916341	+	0.400399i	$0.131128\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.3500	−0.603471	−0.301736	−	0.953392i	$-0.597566\pi$
$647$	−15.3500	−0.603471	−0.301736	+	0.953392i	$0.597566\pi$
$648$	0	0
$649$	−6.11146	−0.239896
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.5902	−1.82322	−0.911608	−	0.411060i	$-0.865159\pi$
$653$	−46.5902	−1.82322	−0.911608	+	0.411060i	$0.865159\pi$
$654$	0	0
$655$	−28.6525	−1.11954
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19.2588	0.750217	0.375108	−	0.926981i	$-0.377606\pi$
$659$	19.2588	0.750217	0.375108	+	0.926981i	$0.377606\pi$
$660$	0	0
$661$	19.2361	0.748196	0.374098	−	0.927389i	$-0.377952\pi$
$661$	19.2361	0.748196	0.374098	+	0.927389i	$0.377952\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.28825	−0.0887344
$666$	0	0
$667$	−24.3607	−0.943249
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.97831	0.385208
$672$	0	0
$673$	28.8328	1.11142	0.555712	−	0.831375i	$-0.312446\pi$
$673$	28.8328	1.11142	0.555712	+	0.831375i	$0.312446\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.3600	−1.47429	−0.737147	−	0.675732i	$-0.763828\pi$
$677$	−38.3600	−1.47429	−0.737147	+	0.675732i	$0.763828\pi$
$678$	0	0
$679$	12.4721	0.478637
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.1676	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$683$	−23.1676	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$684$	0	0
$685$	33.8885	1.29481
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−37.0246	−1.41052
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	29.6197	1.12354
$696$	0	0
$697$	−4.58359	−0.173616
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.9551	1.16916	0.584579	−	0.811337i	$-0.301260\pi$
$701$	30.9551	1.16916	0.584579	+	0.811337i	$0.301260\pi$
$702$	0	0
$703$	4.47214	0.168670
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.70091	0.101578
$708$	0	0
$709$	−14.2918	−0.536740	−0.268370	−	0.963316i	$-0.586485\pi$
$709$	−14.2918	−0.536740	−0.268370	+	0.963316i	$0.586485\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21.8021	0.816495
$714$	0	0
$715$	20.9443	0.783271
$716$	0	0
$717$	0	0
$718$	0	0
$719$	49.9588	1.86315	0.931575	−	0.363549i	$-0.118435\pi$
$719$	49.9588	1.86315	0.931575	+	0.363549i	$0.118435\pi$
$720$	0	0
$721$	−4.94427	−0.184134
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.03321	−0.0755114
$726$	0	0
$727$	−14.4721	−0.536742	−0.268371	−	0.963316i	$-0.586485\pi$
$727$	−14.4721	−0.536742	−0.268371	+	0.963316i	$0.586485\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.82843	−0.104613
$732$	0	0
$733$	−40.8328	−1.50819	−0.754097	−	0.656763i	$-0.771925\pi$
$733$	−40.8328	−1.50819	−0.754097	+	0.656763i	$0.771925\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	43.6042	1.60618
$738$	0	0
$739$	15.1246	0.556368	0.278184	−	0.960528i	$-0.410268\pi$
$739$	15.1246	0.556368	0.278184	+	0.960528i	$0.410268\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.4304	−0.602772	−0.301386	−	0.953502i	$-0.597449\pi$
$743$	−16.4304	−0.602772	−0.301386	+	0.953502i	$0.597449\pi$
$744$	0	0
$745$	−31.4164	−1.15101
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.53244	−0.275229
$750$	0	0
$751$	28.3607	1.03490	0.517448	−	0.855715i	$-0.326882\pi$
$751$	28.3607	1.03490	0.517448	+	0.855715i	$0.326882\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−23.9628	−0.872096
$756$	0	0
$757$	−19.3050	−0.701650	−0.350825	−	0.936441i	$-0.614099\pi$
$757$	−19.3050	−0.701650	−0.350825	+	0.936441i	$0.614099\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	46.7178	1.69352	0.846759	−	0.531977i	$-0.178551\pi$
$761$	46.7178	1.69352	0.846759	+	0.531977i	$0.178551\pi$
$762$	0	0
$763$	1.05573	0.0382199
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.99226	−0.252476
$768$	0	0
$769$	4.11146	0.148263	0.0741315	−	0.997248i	$-0.476382\pi$
$769$	4.11146	0.148263	0.0741315	+	0.997248i	$0.476382\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.2719	1.59235	0.796175	−	0.605067i	$-0.206854\pi$
$773$	44.2719	1.59235	0.796175	+	0.605067i	$0.206854\pi$
$774$	0	0
$775$	1.81966	0.0653641
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.48528	−0.304017
$780$	0	0
$781$	−35.4164	−1.26730
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.7295	−0.490026
$786$	0	0
$787$	12.8754	0.458958	0.229479	−	0.973314i	$-0.426298\pi$
$787$	12.8754	0.458958	0.229479	+	0.973314i	$0.426298\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.2697	0.507371
$792$	0	0
$793$	11.4164	0.405409
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.6522	0.519008	0.259504	−	0.965742i	$-0.416441\pi$
$797$	14.6522	0.519008	0.259504	+	0.965742i	$0.416441\pi$
$798$	0	0
$799$	−2.18034	−0.0771349
$800$	0	0
$801$	0	0
$802$	0	0
$803$	30.9551	1.09238
$804$	0	0
$805$	−6.47214	−0.228113
$806$	0	0
$807$	0	0
$808$	0	0
$809$	33.1158	1.16429	0.582145	−	0.813085i	$-0.302214\pi$
$809$	33.1158	1.16429	0.582145	+	0.813085i	$0.302214\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	34.6088	1.21229
$816$	0	0
$817$	−5.23607	−0.183187
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.73722	0.235131	0.117565	−	0.993065i	$-0.462491\pi$
$821$	6.73722	0.235131	0.117565	+	0.993065i	$0.462491\pi$
$822$	0	0
$823$	−22.1803	−0.773158	−0.386579	−	0.922256i	$-0.626343\pi$
$823$	−22.1803	−0.773158	−0.386579	+	0.922256i	$0.626343\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−18.4335	−0.640995	−0.320498	−	0.947249i	$-0.603850\pi$
$827$	−18.4335	−0.640995	−0.320498	+	0.947249i	$0.603850\pi$
$828$	0	0
$829$	28.4721	0.988878	0.494439	−	0.869212i	$-0.335374\pi$
$829$	28.4721	0.988878	0.494439	+	0.869212i	$0.335374\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.540182	0.0187162
$834$	0	0
$835$	23.4164	0.810358
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.8021	0.752692	0.376346	−	0.926479i	$-0.377180\pi$
$839$	21.8021	0.752692	0.376346	+	0.926479i	$0.377180\pi$
$840$	0	0
$841$	45.1803	1.55794
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.78437	−0.198989
$846$	0	0
$847$	3.00000	0.103081
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12.6491	0.433606
$852$	0	0
$853$	26.3607	0.902572	0.451286	−	0.892379i	$-0.350965\pi$
$853$	26.3607	0.902572	0.451286	+	0.892379i	$0.350965\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.4636	−0.630704	−0.315352	−	0.948975i	$-0.602123\pi$
$857$	−18.4636	−0.630704	−0.315352	+	0.948975i	$0.602123\pi$
$858$	0	0
$859$	−21.3050	−0.726916	−0.363458	−	0.931611i	$-0.618404\pi$
$859$	−21.3050	−0.726916	−0.363458	+	0.931611i	$0.618404\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.4798	−1.54815	−0.774075	−	0.633094i	$-0.781785\pi$
$863$	−45.4798	−1.54815	−0.774075	+	0.633094i	$0.781785\pi$
$864$	0	0
$865$	−9.52786	−0.323957
$866$	0	0
$867$	0	0
$868$	0	0
$869$	29.6197	1.00478
$870$	0	0
$871$	49.8885	1.69041
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.9010	0.368523
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	53.9651	1.81813	0.909064	−	0.416656i	$-0.136798\pi$
$881$	53.9651	1.81813	0.909064	+	0.416656i	$0.136798\pi$
$882$	0	0
$883$	35.0557	1.17972	0.589860	−	0.807506i	$-0.299183\pi$
$883$	35.0557	1.17972	0.589860	+	0.807506i	$0.299183\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−22.3724	−0.751191	−0.375595	−	0.926784i	$-0.622562\pi$
$887$	−22.3724	−0.751191	−0.375595	+	0.926784i	$0.622562\pi$
$888$	0	0
$889$	19.4164	0.651205
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.03631	−0.135070
$894$	0	0
$895$	24.6525	0.824041
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−66.3892	−2.21420
$900$	0	0
$901$	6.18034	0.205897
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.73722	0.223953
$906$	0	0
$907$	−3.05573	−0.101464	−0.0507319	−	0.998712i	$-0.516155\pi$
$907$	−3.05573	−0.101464	−0.0507319	+	0.998712i	$0.516155\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−9.02546	−0.299027	−0.149513	−	0.988760i	$-0.547771\pi$
$911$	−9.02546	−0.299027	−0.149513	+	0.988760i	$0.547771\pi$
$912$	0	0
$913$	22.4721	0.743719
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.5216	0.413499
$918$	0	0
$919$	−28.9443	−0.954783	−0.477392	−	0.878691i	$-0.658418\pi$
$919$	−28.9443	−0.954783	−0.477392	+	0.878691i	$0.658418\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−40.5207	−1.33376
$924$	0	0
$925$	1.05573	0.0347121
$926$	0	0
$927$	0	0
$928$	0	0
$929$	22.3423	0.733026	0.366513	−	0.930413i	$-0.380551\pi$
$929$	22.3423	0.733026	0.366513	+	0.930413i	$0.380551\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.49613	−0.114336
$936$	0	0
$937$	−10.9443	−0.357534	−0.178767	−	0.983891i	$-0.557211\pi$
$937$	−10.9443	−0.357534	−0.178767	+	0.983891i	$0.557211\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.1776	−1.50535	−0.752673	−	0.658395i	$-0.771236\pi$
$941$	−46.1776	−1.50535	−0.752673	+	0.658395i	$0.771236\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.65375	0.118731	0.0593655	−	0.998236i	$-0.481092\pi$
$947$	3.65375	0.118731	0.0593655	+	0.998236i	$0.481092\pi$
$948$	0	0
$949$	35.4164	1.14967
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−13.8570	−0.448872	−0.224436	−	0.974489i	$-0.572054\pi$
$953$	−13.8570	−0.448872	−0.224436	+	0.974489i	$0.572054\pi$
$954$	0	0
$955$	30.4721	0.986055
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−14.8098	−0.478235
$960$	0	0
$961$	28.4164	0.916658
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−41.1884	−1.32590
$966$	0	0
$967$	−50.5410	−1.62529	−0.812645	−	0.582759i	$-0.801973\pi$
$967$	−50.5410	−1.62529	−0.812645	+	0.582759i	$0.801973\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.3137	0.363074	0.181537	−	0.983384i	$-0.441893\pi$
$971$	11.3137	0.363074	0.181537	+	0.983384i	$0.441893\pi$
$972$	0	0
$973$	−12.9443	−0.414974
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0.952843	0.0304842	0.0152421	−	0.999884i	$-0.495148\pi$
$977$	0.952843	0.0304842	0.0152421	+	0.999884i	$0.495148\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	0	0
$982$	0	0
$983$	29.6197	0.944721	0.472360	−	0.881405i	$-0.343402\pi$
$983$	29.6197	0.944721	0.472360	+	0.881405i	$0.343402\pi$
$984$	0	0
$985$	46.8328	1.49222
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.8098	−0.470925
$990$	0	0
$991$	−29.8885	−0.949441	−0.474720	−	0.880137i	$-0.657451\pi$
$991$	−29.8885	−0.949441	−0.474720	+	0.880137i	$0.657451\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.3060	0.580338
$996$	0	0
$997$	53.7771	1.70314	0.851569	−	0.524243i	$-0.175652\pi$
$997$	53.7771	1.70314	0.851569	+	0.524243i	$0.175652\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.q.1.1	✓	4	3.2	odd	2	inner
4788.2.a.q.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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.28825 q^{5} -1.00000 q^{7} -2.82843 q^{11} -3.23607 q^{13} +0.540182 q^{17} +1.00000 q^{19} +2.82843 q^{23} +0.236068 q^{25} -8.61280 q^{29} +7.70820 q^{31} -2.28825 q^{35} +4.47214 q^{37} -8.48528 q^{41} -5.23607 q^{43} -4.03631 q^{47} +1.00000 q^{49} +11.4412 q^{53} -6.47214 q^{55} +2.16073 q^{59} -3.52786 q^{61} -7.40492 q^{65} -15.4164 q^{67} +12.5216 q^{71} -10.9443 q^{73} +2.82843 q^{77} -10.4721 q^{79} -7.94510 q^{83} +1.23607 q^{85} -8.48528 q^{89} +3.23607 q^{91} +2.28825 q^{95} -12.4721 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{13} + 4 q^{19} - 8 q^{25} + 4 q^{31} - 12 q^{43} + 4 q^{49} - 8 q^{55} - 32 q^{61} - 8 q^{67} - 8 q^{73} - 24 q^{79} - 4 q^{85} + 4 q^{91} - 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^13 + 4 * q^19 - 8 * q^25 + 4 * q^31 - 12 * q^43 + 4 * q^49 - 8 * q^55 - 32 * q^61 - 8 * q^67 - 8 * q^73 - 24 * q^79 - 4 * q^85 + 4 * q^91 - 32 * q^97

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