LMFDB - Embedded newform 4788.2.a.t.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)

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")

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);

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 = [6,0,0,0,0,0,-6,0,0,0,0,0,4] 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:	$0$
Dimension:	$6$
Coefficient field:	6.6.56310016.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 14x^{2} - 4 $ x^6 - 9x^4 + 14x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.66648$ 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)

$f(q)$	$=$	$q+1.79367 q^{5} -1.00000 q^{7} +3.83284 q^{11} -3.43742 q^{13} +0.293558 q^{17} -1.00000 q^{19} +6.83306 q^{23} -1.78276 q^{25} +5.62651 q^{29} -4.78276 q^{31} -1.79367 q^{35} +11.5655 q^{37} -5.57846 q^{41} +4.78276 q^{43} -12.7051 q^{47} +1.00000 q^{49} +4.12640 q^{53} +6.87484 q^{55} +11.5655 q^{61} -6.16558 q^{65} +10.2202 q^{67} +3.53929 q^{71} -10.4404 q^{73} -3.83284 q^{77} -2.22018 q^{79} +17.7925 q^{83} +0.526544 q^{85} +5.08744 q^{89} +3.43742 q^{91} -1.79367 q^{95} +9.52949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} + 4 q^{13} - 6 q^{19} + 14 q^{25} - 4 q^{31} + 20 q^{37} + 4 q^{43} + 6 q^{49} - 8 q^{55} + 20 q^{61} + 12 q^{67} + 36 q^{73} + 36 q^{79} + 28 q^{85} - 4 q^{91} + 8 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 + 4 * q^13 - 6 * q^19 + 14 * q^25 - 4 * q^31 + 20 * q^37 + 4 * q^43 + 6 * q^49 - 8 * q^55 + 20 * q^61 + 12 * q^67 + 36 * q^73 + 36 * q^79 + 28 * q^85 - 4 * q^91 + 8 * 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
$5$	1.79367	0.802152	0.401076	−	0.916045i	$-0.368636\pi$
$5$	1.79367	0.802152	0.401076	+	0.916045i	$0.368636\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.83284	1.15565	0.577823	−	0.816162i	$-0.303902\pi$
$11$	3.83284	1.15565	0.577823	+	0.816162i	$0.303902\pi$
$12$	0	0
$13$	−3.43742	−0.953369	−0.476684	−	0.879075i	$-0.658162\pi$
$13$	−3.43742	−0.953369	−0.476684	+	0.879075i	$0.658162\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.293558	0.0711982	0.0355991	−	0.999366i	$-0.488666\pi$
$17$	0.293558	0.0711982	0.0355991	+	0.999366i	$0.488666\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.83306	1.42479	0.712396	−	0.701778i	$-0.247610\pi$
$23$	6.83306	1.42479	0.712396	+	0.701778i	$0.247610\pi$
$24$	0	0
$25$	−1.78276	−0.356553
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.62651	1.04482	0.522408	−	0.852696i	$-0.325034\pi$
$29$	5.62651	1.04482	0.522408	+	0.852696i	$0.325034\pi$
$30$	0	0
$31$	−4.78276	−0.859010	−0.429505	−	0.903065i	$-0.641312\pi$
$31$	−4.78276	−0.859010	−0.429505	+	0.903065i	$0.641312\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.79367	−0.303185
$36$	0	0
$37$	11.5655	1.90136	0.950681	−	0.310171i	$-0.100386\pi$
$37$	11.5655	1.90136	0.950681	+	0.310171i	$0.100386\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.57846	−0.871210	−0.435605	−	0.900138i	$-0.643465\pi$
$41$	−5.57846	−0.871210	−0.435605	+	0.900138i	$0.643465\pi$
$42$	0	0
$43$	4.78276	0.729365	0.364682	−	0.931132i	$-0.381178\pi$
$43$	4.78276	0.729365	0.364682	+	0.931132i	$0.381178\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.7051	−1.85323	−0.926613	−	0.376016i	$-0.877294\pi$
$47$	−12.7051	−1.85323	−0.926613	+	0.376016i	$0.877294\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.12640	0.566805	0.283402	−	0.959001i	$-0.408537\pi$
$53$	4.12640	0.566805	0.283402	+	0.959001i	$0.408537\pi$
$54$	0	0
$55$	6.87484	0.927003
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	11.5655	1.48081	0.740407	−	0.672159i	$-0.234633\pi$
$61$	11.5655	1.48081	0.740407	+	0.672159i	$0.234633\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.16558	−0.764746
$66$	0	0
$67$	10.2202	1.24859	0.624297	−	0.781187i	$-0.285385\pi$
$67$	10.2202	1.24859	0.624297	+	0.781187i	$0.285385\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.53929	0.420036	0.210018	−	0.977698i	$-0.432648\pi$
$71$	3.53929	0.420036	0.210018	+	0.977698i	$0.432648\pi$
$72$	0	0
$73$	−10.4404	−1.22195	−0.610976	−	0.791649i	$-0.709223\pi$
$73$	−10.4404	−1.22195	−0.610976	+	0.791649i	$0.709223\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.83284	−0.436793
$78$	0	0
$79$	−2.22018	−0.249790	−0.124895	−	0.992170i	$-0.539859\pi$
$79$	−2.22018	−0.249790	−0.124895	+	0.992170i	$0.539859\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.7925	1.95298	0.976491	−	0.215557i	$-0.0691566\pi$
$83$	17.7925	1.95298	0.976491	+	0.215557i	$0.0691566\pi$
$84$	0	0
$85$	0.526544	0.0571118
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.08744	0.539267	0.269634	−	0.962963i	$-0.413097\pi$
$89$	5.08744	0.539267	0.269634	+	0.962963i	$0.413097\pi$
$90$	0	0
$91$	3.43742	0.360339
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.79367	−0.184026
$96$	0	0
$97$	9.52949	0.967573	0.483787	−	0.875186i	$-0.339261\pi$
$97$	9.52949	0.967573	0.483787	+	0.875186i	$0.339261\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9595	−1.09051	−0.545254	−	0.838271i	$-0.683567\pi$
$101$	−10.9595	−1.09051	−0.545254	+	0.838271i	$0.683567\pi$
$102$	0	0
$103$	−12.4404	−1.22579	−0.612893	−	0.790166i	$-0.709994\pi$
$103$	−12.4404	−1.22579	−0.612893	+	0.790166i	$0.709994\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.54815	−0.149666	−0.0748328	−	0.997196i	$-0.523842\pi$
$107$	−1.54815	−0.149666	−0.0748328	+	0.997196i	$0.523842\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.2926	1.81490	0.907449	−	0.420163i	$-0.138027\pi$
$113$	19.2926	1.81490	0.907449	+	0.420163i	$0.138027\pi$
$114$	0	0
$115$	12.2562	1.14290
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.293558	−0.0269104
$120$	0	0
$121$	3.69069	0.335517
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1660	−1.08816
$126$	0	0
$127$	7.52949	0.668134	0.334067	−	0.942549i	$-0.391579\pi$
$127$	7.52949	0.668134	0.334067	+	0.942549i	$0.391579\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.7139	0.936082	0.468041	−	0.883707i	$-0.344960\pi$
$131$	10.7139	0.936082	0.468041	+	0.883707i	$0.344960\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.3316	−1.56617	−0.783087	−	0.621913i	$-0.786356\pi$
$137$	−18.3316	−1.56617	−0.783087	+	0.621913i	$0.786356\pi$
$138$	0	0
$139$	−1.30931	−0.111054	−0.0555271	−	0.998457i	$-0.517684\pi$
$139$	−1.30931	−0.111054	−0.0555271	+	0.998457i	$0.517684\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.1751	−1.10176
$144$	0	0
$145$	10.0921	0.838101
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.07857	0.579899	0.289950	−	0.957042i	$-0.406361\pi$
$149$	7.07857	0.579899	0.289950	+	0.957042i	$0.406361\pi$
$150$	0	0
$151$	19.5295	1.58929	0.794644	−	0.607076i	$-0.207658\pi$
$151$	19.5295	1.58929	0.794644	+	0.607076i	$0.207658\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.57868	−0.689056
$156$	0	0
$157$	−10.4404	−0.833232	−0.416616	−	0.909083i	$-0.636784\pi$
$157$	−10.4404	−0.833232	−0.416616	+	0.909083i	$0.636784\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.83306	−0.538520
$162$	0	0
$163$	−3.47346	−0.272062	−0.136031	−	0.990705i	$-0.543435\pi$
$163$	−3.47346	−0.272062	−0.136031	+	0.990705i	$0.543435\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.0788	−0.779920	−0.389960	−	0.920832i	$-0.627511\pi$
$167$	−10.0788	−0.779920	−0.389960	+	0.920832i	$0.627511\pi$
$168$	0	0
$169$	−1.18415	−0.0910883
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.08744	0.386791	0.193395	−	0.981121i	$-0.438050\pi$
$173$	5.08744	0.386791	0.193395	+	0.981121i	$0.438050\pi$
$174$	0	0
$175$	1.78276	0.134764
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.8012	−0.956804	−0.478402	−	0.878141i	$-0.658784\pi$
$179$	−12.8012	−0.956804	−0.478402	+	0.878141i	$0.658784\pi$
$180$	0	0
$181$	4.03604	0.299996	0.149998	−	0.988686i	$-0.452073\pi$
$181$	4.03604	0.299996	0.149998	+	0.988686i	$0.452073\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.7447	1.52518
$186$	0	0
$187$	1.12516	0.0822799
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.6736	−1.78532	−0.892660	−	0.450730i	$-0.851164\pi$
$191$	−24.6736	−1.78532	−0.892660	+	0.450730i	$0.851164\pi$
$192$	0	0
$193$	25.1311	1.80897	0.904487	−	0.426502i	$-0.140254\pi$
$193$	25.1311	1.80897	0.904487	+	0.426502i	$0.140254\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.3314	1.09232	0.546158	−	0.837682i	$-0.316090\pi$
$197$	15.3314	1.09232	0.546158	+	0.837682i	$0.316090\pi$
$198$	0	0
$199$	19.1311	1.35616	0.678082	−	0.734986i	$-0.262811\pi$
$199$	19.1311	1.35616	0.678082	+	0.734986i	$0.262811\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.62651	−0.394904
$204$	0	0
$205$	−10.0059	−0.698842
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.83284	−0.265123
$210$	0	0
$211$	−11.5295	−0.793723	−0.396862	−	0.917878i	$-0.629901\pi$
$211$	−11.5295	−0.793723	−0.396862	+	0.917878i	$0.629901\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.57868	0.585061
$216$	0	0
$217$	4.78276	0.324675
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.00908	−0.0678782
$222$	0	0
$223$	−6.34829	−0.425113	−0.212557	−	0.977149i	$-0.568179\pi$
$223$	−6.34829	−0.425113	−0.212557	+	0.977149i	$0.568179\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.4099	1.48740	0.743700	−	0.668513i	$-0.233069\pi$
$227$	22.4099	1.48740	0.743700	+	0.668513i	$0.233069\pi$
$228$	0	0
$229$	20.8748	1.37945	0.689724	−	0.724072i	$-0.257732\pi$
$229$	20.8748	1.37945	0.689724	+	0.724072i	$0.257732\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.4277	−1.20724	−0.603619	−	0.797273i	$-0.706275\pi$
$233$	−18.4277	−1.20724	−0.603619	+	0.797273i	$0.706275\pi$
$234$	0	0
$235$	−22.7887	−1.48657
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.4206	−0.868107	−0.434053	−	0.900887i	$-0.642917\pi$
$239$	−13.4206	−0.868107	−0.434053	+	0.900887i	$0.642917\pi$
$240$	0	0
$241$	15.0950	0.972356	0.486178	−	0.873860i	$-0.338391\pi$
$241$	15.0950	0.972356	0.486178	+	0.873860i	$0.338391\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.79367	0.114593
$246$	0	0
$247$	3.43742	0.218718
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.6269	0.733886	0.366943	−	0.930243i	$-0.380404\pi$
$251$	11.6269	0.733886	0.366943	+	0.930243i	$0.380404\pi$
$252$	0	0
$253$	26.1900	1.64655
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.0059	1.31031	0.655157	−	0.755493i	$-0.272603\pi$
$257$	21.0059	1.31031	0.655157	+	0.755493i	$0.272603\pi$
$258$	0	0
$259$	−11.5655	−0.718647
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.92893	−0.242268	−0.121134	−	0.992636i	$-0.538653\pi$
$263$	−3.92893	−0.242268	−0.121134	+	0.992636i	$0.538653\pi$
$264$	0	0
$265$	7.40138	0.454663
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.8317	1.20916	0.604580	−	0.796544i	$-0.293341\pi$
$269$	19.8317	1.20916	0.604580	+	0.796544i	$0.293341\pi$
$270$	0	0
$271$	2.69069	0.163448	0.0817239	−	0.996655i	$-0.473957\pi$
$271$	2.69069	0.163448	0.0817239	+	0.996655i	$0.473957\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.83306	−0.412049
$276$	0	0
$277$	11.9079	0.715478	0.357739	−	0.933822i	$-0.383548\pi$
$277$	11.9079	0.715478	0.357739	+	0.933822i	$0.383548\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.8886	1.06714	0.533572	−	0.845754i	$-0.320849\pi$
$281$	17.8886	1.06714	0.533572	+	0.845754i	$0.320849\pi$
$282$	0	0
$283$	−21.5655	−1.28194	−0.640969	−	0.767567i	$-0.721467\pi$
$283$	−21.5655	−1.28194	−0.640969	+	0.767567i	$0.721467\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.57846	0.329286
$288$	0	0
$289$	−16.9138	−0.994931
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.75291	−0.569771	−0.284886	−	0.958562i	$-0.591956\pi$
$293$	−9.75291	−0.569771	−0.284886	+	0.958562i	$0.591956\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−23.4881	−1.35835
$300$	0	0
$301$	−4.78276	−0.275674
$302$	0	0
$303$	0	0
$304$	0	0
$305$	20.7447	1.18784
$306$	0	0
$307$	−32.0980	−1.83193	−0.915964	−	0.401260i	$-0.868572\pi$
$307$	−32.0980	−1.83193	−0.915964	+	0.401260i	$0.868572\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−7.12662	−0.404113	−0.202057	−	0.979374i	$-0.564763\pi$
$311$	−7.12662	−0.404113	−0.202057	+	0.979374i	$0.564763\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.864947	−0.0485803	−0.0242901	−	0.999705i	$-0.507733\pi$
$317$	−0.864947	−0.0485803	−0.0242901	+	0.999705i	$0.507733\pi$
$318$	0	0
$319$	21.5655	1.20744
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.293558	−0.0163340
$324$	0	0
$325$	6.12811	0.339926
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.7051	0.700454
$330$	0	0
$331$	15.7857	0.867661	0.433831	−	0.900994i	$-0.357162\pi$
$331$	15.7857	0.867661	0.433831	+	0.900994i	$0.357162\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	18.3316	1.00156
$336$	0	0
$337$	18.4404	1.00451	0.502255	−	0.864719i	$-0.332504\pi$
$337$	18.4404	1.00451	0.502255	+	0.864719i	$0.332504\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−18.3316	−0.992711
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−33.4175	−1.79394	−0.896971	−	0.442089i	$-0.854238\pi$
$347$	−33.4175	−1.79394	−0.896971	+	0.442089i	$0.854238\pi$
$348$	0	0
$349$	8.87484	0.475059	0.237530	−	0.971380i	$-0.423662\pi$
$349$	8.87484	0.475059	0.237530	+	0.971380i	$0.423662\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−31.9654	−1.70135	−0.850673	−	0.525695i	$-0.823805\pi$
$353$	−31.9654	−1.70135	−0.850673	+	0.525695i	$0.823805\pi$
$354$	0	0
$355$	6.34829	0.336932
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.00729	−0.422609	−0.211304	−	0.977420i	$-0.567771\pi$
$359$	−8.00729	−0.422609	−0.211304	+	0.977420i	$0.567771\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18.7265	−0.980191
$366$	0	0
$367$	4.18415	0.218411	0.109205	−	0.994019i	$-0.465169\pi$
$367$	4.18415	0.218411	0.109205	+	0.994019i	$0.465169\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.12640	−0.214232
$372$	0	0
$373$	29.5714	1.53115	0.765575	−	0.643346i	$-0.222454\pi$
$373$	29.5714	1.53115	0.765575	+	0.643346i	$0.222454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.3407	−0.996095
$378$	0	0
$379$	−7.52949	−0.386764	−0.193382	−	0.981124i	$-0.561946\pi$
$379$	−7.52949	−0.386764	−0.193382	+	0.981124i	$0.561946\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	25.6023	1.30822	0.654109	−	0.756400i	$-0.273044\pi$
$383$	25.6023	1.30822	0.654109	+	0.756400i	$0.273044\pi$
$384$	0	0
$385$	−6.87484	−0.350374
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.0756	1.52490	0.762448	−	0.647050i	$-0.223998\pi$
$389$	30.0756	1.52490	0.762448	+	0.647050i	$0.223998\pi$
$390$	0	0
$391$	2.00590	0.101443
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3.98227	−0.200370
$396$	0	0
$397$	21.0590	1.05692	0.528460	−	0.848958i	$-0.322770\pi$
$397$	21.0590	1.05692	0.528460	+	0.848958i	$0.322770\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.2052	−0.709373	−0.354687	−	0.934985i	$-0.615412\pi$
$401$	−14.2052	−0.709373	−0.354687	+	0.934985i	$0.615412\pi$
$402$	0	0
$403$	16.4404	0.818953
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.3289	2.19730
$408$	0	0
$409$	−14.1281	−0.698590	−0.349295	−	0.937013i	$-0.613579\pi$
$409$	−14.1281	−0.698590	−0.349295	+	0.937013i	$0.613579\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	31.9138	1.56659
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.72281	0.426137	0.213069	−	0.977037i	$-0.431654\pi$
$419$	8.72281	0.426137	0.213069	+	0.977037i	$0.431654\pi$
$420$	0	0
$421$	−33.5714	−1.63617	−0.818086	−	0.575096i	$-0.804965\pi$
$421$	−33.5714	−1.63617	−0.818086	+	0.575096i	$0.804965\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−0.523345	−0.0253859
$426$	0	0
$427$	−11.5655	−0.559695
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.96125	0.190807	0.0954034	−	0.995439i	$-0.469586\pi$
$431$	3.96125	0.190807	0.0954034	+	0.995439i	$0.469586\pi$
$432$	0	0
$433$	23.1671	1.11334	0.556670	−	0.830734i	$-0.312079\pi$
$433$	23.1671	1.11334	0.556670	+	0.830734i	$0.312079\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.83306	−0.326870
$438$	0	0
$439$	−5.03899	−0.240498	−0.120249	−	0.992744i	$-0.538369\pi$
$439$	−5.03899	−0.240498	−0.120249	+	0.992744i	$0.538369\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.91099	0.233328	0.116664	−	0.993171i	$-0.462780\pi$
$443$	4.91099	0.233328	0.116664	+	0.993171i	$0.462780\pi$
$444$	0	0
$445$	9.12516	0.432574
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−35.0460	−1.65392	−0.826961	−	0.562260i	$-0.809932\pi$
$449$	−35.0460	−1.65392	−0.826961	+	0.562260i	$0.809932\pi$
$450$	0	0
$451$	−21.3814	−1.00681
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.16558	0.289047
$456$	0	0
$457$	−0.348294	−0.0162925	−0.00814626	−	0.999967i	$-0.502593\pi$
$457$	−0.348294	−0.0162925	−0.00814626	+	0.999967i	$0.502593\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.5291	0.862984	0.431492	−	0.902117i	$-0.357987\pi$
$461$	18.5291	0.862984	0.431492	+	0.902117i	$0.357987\pi$
$462$	0	0
$463$	−39.4994	−1.83569	−0.917845	−	0.396938i	$-0.870073\pi$
$463$	−39.4994	−1.83569	−0.917845	+	0.396938i	$0.870073\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.2054	0.796171	0.398086	−	0.917348i	$-0.369675\pi$
$467$	17.2054	0.796171	0.398086	+	0.917348i	$0.369675\pi$
$468$	0	0
$469$	−10.2202	−0.471924
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18.3316	0.842887
$474$	0	0
$475$	1.78276	0.0817989
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.7834	−0.766855	−0.383427	−	0.923571i	$-0.625256\pi$
$479$	−16.7834	−0.766855	−0.383427	+	0.923571i	$0.625256\pi$
$480$	0	0
$481$	−39.7556	−1.81270
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.0927	0.776141
$486$	0	0
$487$	3.34535	0.151592	0.0757960	−	0.997123i	$-0.475850\pi$
$487$	3.34535	0.151592	0.0757960	+	0.997123i	$0.475850\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−25.0686	−1.13133	−0.565664	−	0.824636i	$-0.691380\pi$
$491$	−25.0686	−1.13133	−0.565664	+	0.824636i	$0.691380\pi$
$492$	0	0
$493$	1.65171	0.0743891
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.53929	−0.158759
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.2983	−1.66305	−0.831525	−	0.555487i	$-0.812532\pi$
$503$	−37.2983	−1.66305	−0.831525	+	0.555487i	$0.812532\pi$
$504$	0	0
$505$	−19.6576	−0.874752
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.5149	0.909307	0.454654	−	0.890668i	$-0.349763\pi$
$509$	20.5149	0.909307	0.454654	+	0.890668i	$0.349763\pi$
$510$	0	0
$511$	10.4404	0.461855
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−22.3139	−0.983266
$516$	0	0
$517$	−48.6966	−2.14167
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.0057	−0.788845	−0.394422	−	0.918929i	$-0.629055\pi$
$521$	−18.0057	−0.788845	−0.394422	+	0.918929i	$0.629055\pi$
$522$	0	0
$523$	−30.7887	−1.34629	−0.673147	−	0.739509i	$-0.735058\pi$
$523$	−30.7887	−1.34629	−0.673147	+	0.739509i	$0.735058\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.40402	−0.0611600
$528$	0	0
$529$	23.6907	1.03003
$530$	0	0
$531$	0	0
$532$	0	0
$533$	19.1755	0.830584
$534$	0	0
$535$	−2.77687	−0.120054
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.83284	0.165092
$540$	0	0
$541$	37.3152	1.60431	0.802153	−	0.597119i	$-0.203688\pi$
$541$	37.3152	1.60431	0.802153	+	0.597119i	$0.203688\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	17.9367	0.768322
$546$	0	0
$547$	34.6606	1.48198	0.740989	−	0.671517i	$-0.234357\pi$
$547$	34.6606	1.48198	0.740989	+	0.671517i	$0.234357\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.62651	−0.239697
$552$	0	0
$553$	2.22018	0.0944118
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.56938	0.193611	0.0968055	−	0.995303i	$-0.469138\pi$
$557$	4.56938	0.193611	0.0968055	+	0.995303i	$0.469138\pi$
$558$	0	0
$559$	−16.4404	−0.695353
$560$	0	0
$561$	0	0
$562$	0	0
$563$	35.5850	1.49973	0.749865	−	0.661591i	$-0.230118\pi$
$563$	35.5850	1.49973	0.749865	+	0.661591i	$0.230118\pi$
$564$	0	0
$565$	34.6045	1.45582
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.3619	0.937460	0.468730	−	0.883342i	$-0.344712\pi$
$569$	22.3619	0.937460	0.468730	+	0.883342i	$0.344712\pi$
$570$	0	0
$571$	−27.5714	−1.15383	−0.576914	−	0.816805i	$-0.695743\pi$
$571$	−27.5714	−1.15383	−0.576914	+	0.816805i	$0.695743\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.1817	−0.508014
$576$	0	0
$577$	36.2621	1.50961	0.754806	−	0.655948i	$-0.227731\pi$
$577$	36.2621	1.50961	0.754806	+	0.655948i	$0.227731\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.7925	−0.738158
$582$	0	0
$583$	15.8159	0.655026
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.1330	1.40882	0.704409	−	0.709794i	$-0.251212\pi$
$587$	34.1330	1.40882	0.704409	+	0.709794i	$0.251212\pi$
$588$	0	0
$589$	4.78276	0.197070
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.9656	1.43587	0.717933	−	0.696113i	$-0.245089\pi$
$593$	34.9656	1.43587	0.717933	+	0.696113i	$0.245089\pi$
$594$	0	0
$595$	−0.526544	−0.0215862
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.7057	0.886872	0.443436	−	0.896306i	$-0.353759\pi$
$599$	21.7057	0.886872	0.443436	+	0.896306i	$0.353759\pi$
$600$	0	0
$601$	−24.2202	−0.987962	−0.493981	−	0.869473i	$-0.664459\pi$
$601$	−24.2202	−0.987962	−0.493981	+	0.869473i	$0.664459\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.61986	0.269136
$606$	0	0
$607$	36.8807	1.49694	0.748471	−	0.663167i	$-0.230788\pi$
$607$	36.8807	1.49694	0.748471	+	0.663167i	$0.230788\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	43.6727	1.76681
$612$	0	0
$613$	4.77687	0.192936	0.0964679	−	0.995336i	$-0.469245\pi$
$613$	4.77687	0.192936	0.0964679	+	0.995336i	$0.469245\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.5060	−0.906059	−0.453029	−	0.891496i	$-0.649657\pi$
$617$	−22.5060	−0.906059	−0.453029	+	0.891496i	$0.649657\pi$
$618$	0	0
$619$	−36.4404	−1.46466	−0.732331	−	0.680949i	$-0.761568\pi$
$619$	−36.4404	−1.46466	−0.732331	+	0.680949i	$0.761568\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.08744	−0.203824
$624$	0	0
$625$	−12.9079	−0.516317
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.39515	0.135374
$630$	0	0
$631$	−35.6576	−1.41951	−0.709753	−	0.704450i	$-0.751194\pi$
$631$	−35.6576	−1.41951	−0.709753	+	0.704450i	$0.751194\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.5054	0.535945
$636$	0	0
$637$	−3.43742	−0.136196
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.6246	−1.72307	−0.861535	−	0.507698i	$-0.830496\pi$
$641$	−43.6246	−1.72307	−0.861535	+	0.507698i	$0.830496\pi$
$642$	0	0
$643$	−21.5655	−0.850461	−0.425231	−	0.905085i	$-0.639807\pi$
$643$	−21.5655	−0.850461	−0.425231	+	0.905085i	$0.639807\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.7059	−0.971291	−0.485645	−	0.874156i	$-0.661415\pi$
$647$	−24.7059	−0.971291	−0.485645	+	0.874156i	$0.661415\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.3312	−0.482556	−0.241278	−	0.970456i	$-0.577566\pi$
$653$	−12.3312	−0.482556	−0.241278	+	0.970456i	$0.577566\pi$
$654$	0	0
$655$	19.2172	0.750880
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.9672	0.972583	0.486292	−	0.873797i	$-0.338349\pi$
$659$	24.9672	0.972583	0.486292	+	0.873797i	$0.338349\pi$
$660$	0	0
$661$	−34.8247	−1.35452	−0.677262	−	0.735742i	$-0.736834\pi$
$661$	−34.8247	−1.35452	−0.677262	+	0.735742i	$0.736834\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.79367	0.0695554
$666$	0	0
$667$	38.4463	1.48865
$668$	0	0
$669$	0	0
$670$	0	0
$671$	44.3289	1.71130
$672$	0	0
$673$	26.4404	1.01920	0.509601	−	0.860411i	$-0.329793\pi$
$673$	26.4404	1.01920	0.509601	+	0.860411i	$0.329793\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.1453	1.50448	0.752239	−	0.658891i	$-0.228974\pi$
$677$	39.1453	1.50448	0.752239	+	0.658891i	$0.228974\pi$
$678$	0	0
$679$	−9.52949	−0.365708
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−47.1159	−1.80284	−0.901420	−	0.432946i	$-0.857474\pi$
$683$	−47.1159	−1.80284	−0.901420	+	0.432946i	$0.857474\pi$
$684$	0	0
$685$	−32.8807	−1.25631
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14.1842	−0.540374
$690$	0	0
$691$	15.5714	0.592365	0.296183	−	0.955131i	$-0.404286\pi$
$691$	15.5714	0.592365	0.296183	+	0.955131i	$0.404286\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.34846	−0.0890823
$696$	0	0
$697$	−1.63760	−0.0620286
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.3312	−0.465741	−0.232871	−	0.972508i	$-0.574812\pi$
$701$	−12.3312	−0.465741	−0.232871	+	0.972508i	$0.574812\pi$
$702$	0	0
$703$	−11.5655	−0.436202
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.9595	0.412173
$708$	0	0
$709$	−22.9669	−0.862541	−0.431270	−	0.902223i	$-0.641934\pi$
$709$	−22.9669	−0.862541	−0.431270	+	0.902223i	$0.641934\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−32.6809	−1.22391
$714$	0	0
$715$	−23.6317	−0.883776
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14.6001	0.544493	0.272246	−	0.962228i	$-0.412233\pi$
$719$	14.6001	0.544493	0.272246	+	0.962228i	$0.412233\pi$
$720$	0	0
$721$	12.4404	0.463304
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.0307	−0.372532
$726$	0	0
$727$	10.0059	0.371098	0.185549	−	0.982635i	$-0.440594\pi$
$727$	10.0059	0.371098	0.185549	+	0.982635i	$0.440594\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.40402	0.0519295
$732$	0	0
$733$	30.8807	1.14061	0.570303	−	0.821434i	$-0.306826\pi$
$733$	30.8807	1.14061	0.570303	+	0.821434i	$0.306826\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	39.1724	1.44293
$738$	0	0
$739$	−7.40138	−0.272264	−0.136132	−	0.990691i	$-0.543467\pi$
$739$	−7.40138	−0.272264	−0.136132	+	0.990691i	$0.543467\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24.5452	−0.900477	−0.450238	−	0.892908i	$-0.648661\pi$
$743$	−24.5452	−0.900477	−0.450238	+	0.892908i	$0.648661\pi$
$744$	0	0
$745$	12.6966	0.465167
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.54815	0.0565683
$750$	0	0
$751$	−44.0419	−1.60711	−0.803556	−	0.595228i	$-0.797062\pi$
$751$	−44.0419	−1.60711	−0.803556	+	0.595228i	$0.797062\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	35.0294	1.27485
$756$	0	0
$757$	−7.49346	−0.272354	−0.136177	−	0.990685i	$-0.543482\pi$
$757$	−7.49346	−0.272354	−0.136177	+	0.990685i	$0.543482\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.3785	−1.24622	−0.623110	−	0.782135i	$-0.714131\pi$
$761$	−34.3785	−1.24622	−0.623110	+	0.782135i	$0.714131\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1.13106	0.0407870	0.0203935	−	0.999792i	$-0.493508\pi$
$769$	1.13106	0.0407870	0.0203935	+	0.999792i	$0.493508\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.06971	−0.326215	−0.163107	−	0.986608i	$-0.552152\pi$
$773$	−9.06971	−0.326215	−0.163107	+	0.986608i	$0.552152\pi$
$774$	0	0
$775$	8.52654	0.306283
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.57846	0.199869
$780$	0	0
$781$	13.5655	0.485413
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.7265	−0.668378
$786$	0	0
$787$	−12.3424	−0.439959	−0.219979	−	0.975505i	$-0.570599\pi$
$787$	−12.3424	−0.439959	−0.219979	+	0.975505i	$0.570599\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19.2926	−0.685967
$792$	0	0
$793$	−39.7556	−1.41176
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.1020	−0.747472	−0.373736	−	0.927535i	$-0.621923\pi$
$797$	−21.1020	−0.747472	−0.373736	+	0.927535i	$0.621923\pi$
$798$	0	0
$799$	−3.72968	−0.131946
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−40.0163	−1.41214
$804$	0	0
$805$	−12.2562	−0.431975
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.6482	0.515002	0.257501	−	0.966278i	$-0.417101\pi$
$809$	14.6482	0.515002	0.257501	+	0.966278i	$0.417101\pi$
$810$	0	0
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.23022	−0.218235
$816$	0	0
$817$	−4.78276	−0.167328
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.7445	−0.619286	−0.309643	−	0.950853i	$-0.600210\pi$
$821$	−17.7445	−0.619286	−0.309643	+	0.950853i	$0.600210\pi$
$822$	0	0
$823$	14.5324	0.506569	0.253285	−	0.967392i	$-0.418489\pi$
$823$	14.5324	0.506569	0.253285	+	0.967392i	$0.418489\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.85608	0.0993157	0.0496578	−	0.998766i	$-0.484187\pi$
$827$	2.85608	0.0993157	0.0496578	+	0.998766i	$0.484187\pi$
$828$	0	0
$829$	15.3512	0.533171	0.266585	−	0.963811i	$-0.414105\pi$
$829$	15.3512	0.533171	0.266585	+	0.963811i	$0.414105\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.293558	0.0101712
$834$	0	0
$835$	−18.0780	−0.625614
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.3141	−0.873939	−0.436969	−	0.899476i	$-0.643948\pi$
$839$	−25.3141	−0.873939	−0.436969	+	0.899476i	$0.643948\pi$
$840$	0	0
$841$	2.65760	0.0916415
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.12397	−0.0730666
$846$	0	0
$847$	−3.69069	−0.126814
$848$	0	0
$849$	0	0
$850$	0	0
$851$	79.0279	2.70904
$852$	0	0
$853$	−0.434470	−0.0148760	−0.00743799	−	0.999972i	$-0.502368\pi$
$853$	−0.434470	−0.0148760	−0.00743799	+	0.999972i	$0.502368\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−40.7686	−1.39263	−0.696314	−	0.717737i	$-0.745178\pi$
$857$	−40.7686	−1.39263	−0.696314	+	0.717737i	$0.745178\pi$
$858$	0	0
$859$	−18.4345	−0.628976	−0.314488	−	0.949261i	$-0.601833\pi$
$859$	−18.4345	−0.628976	−0.314488	+	0.949261i	$0.601833\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.7991	−1.62710	−0.813550	−	0.581495i	$-0.802468\pi$
$863$	−47.7991	−1.62710	−0.813550	+	0.581495i	$0.802468\pi$
$864$	0	0
$865$	9.12516	0.310265
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8.50962	−0.288669
$870$	0	0
$871$	−35.1311	−1.19037
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1660	0.411286
$876$	0	0
$877$	−6.86894	−0.231948	−0.115974	−	0.993252i	$-0.536999\pi$
$877$	−6.86894	−0.231948	−0.115974	+	0.993252i	$0.536999\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−15.4598	−0.520853	−0.260427	−	0.965494i	$-0.583863\pi$
$881$	−15.4598	−0.520853	−0.260427	+	0.965494i	$0.583863\pi$
$882$	0	0
$883$	−29.7497	−1.00116	−0.500578	−	0.865691i	$-0.666879\pi$
$883$	−29.7497	−1.00116	−0.500578	+	0.865691i	$0.666879\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.2353	0.511551	0.255775	−	0.966736i	$-0.417669\pi$
$887$	15.2353	0.511551	0.255775	+	0.966736i	$0.417669\pi$
$888$	0	0
$889$	−7.52949	−0.252531
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.7051	0.425159
$894$	0	0
$895$	−22.9610	−0.767502
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−26.9103	−0.897508
$900$	0	0
$901$	1.21134	0.0403555
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.23930	0.240642
$906$	0	0
$907$	48.2261	1.60132	0.800660	−	0.599118i	$-0.204482\pi$
$907$	48.2261	1.60132	0.800660	+	0.599118i	$0.204482\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.3801	−0.807748	−0.403874	−	0.914815i	$-0.632337\pi$
$911$	−24.3801	−0.807748	−0.403874	+	0.914815i	$0.632337\pi$
$912$	0	0
$913$	68.1959	2.25696
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.7139	−0.353806
$918$	0	0
$919$	16.3683	0.539940	0.269970	−	0.962869i	$-0.412986\pi$
$919$	16.3683	0.539940	0.269970	+	0.962869i	$0.412986\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−12.1660	−0.400449
$924$	0	0
$925$	−20.6186	−0.677936
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−24.0385	−0.788676	−0.394338	−	0.918965i	$-0.629026\pi$
$929$	−24.0385	−0.788676	−0.394338	+	0.918965i	$0.629026\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.01816	0.0660010
$936$	0	0
$937$	−39.8217	−1.30092	−0.650460	−	0.759541i	$-0.725424\pi$
$937$	−39.8217	−1.30092	−0.650460	+	0.759541i	$0.725424\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.816902	0.0266303	0.0133151	−	0.999911i	$-0.495762\pi$
$941$	0.816902	0.0266303	0.0133151	+	0.999911i	$0.495762\pi$
$942$	0	0
$943$	−38.1180	−1.24129
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16.6550	0.541216	0.270608	−	0.962690i	$-0.412775\pi$
$947$	16.6550	0.541216	0.270608	+	0.962690i	$0.412775\pi$
$948$	0	0
$949$	35.8879	1.16497
$950$	0	0
$951$	0	0
$952$	0	0
$953$	7.61764	0.246760	0.123380	−	0.992360i	$-0.460627\pi$
$953$	7.61764	0.246760	0.123380	+	0.992360i	$0.460627\pi$
$954$	0	0
$955$	−44.2562	−1.43210
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.3316	0.591958
$960$	0	0
$961$	−8.12516	−0.262102
$962$	0	0
$963$	0	0
$964$	0	0
$965$	45.0767	1.45107
$966$	0	0
$967$	−50.9728	−1.63918	−0.819588	−	0.572954i	$-0.805797\pi$
$967$	−50.9728	−1.63918	−0.819588	+	0.572954i	$0.805797\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−43.8378	−1.40682	−0.703412	−	0.710783i	$-0.748341\pi$
$971$	−43.8378	−1.40682	−0.703412	+	0.710783i	$0.748341\pi$
$972$	0	0
$973$	1.30931	0.0419745
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53.2168	−1.70256	−0.851279	−	0.524714i	$-0.824172\pi$
$977$	−53.2168	−1.70256	−0.851279	+	0.524714i	$0.824172\pi$
$978$	0	0
$979$	19.4994	0.623202
$980$	0	0
$981$	0	0
$982$	0	0
$983$	56.8208	1.81230	0.906150	−	0.422956i	$-0.139007\pi$
$983$	56.8208	1.81230	0.906150	+	0.422956i	$0.139007\pi$
$984$	0	0
$985$	27.4994	0.876202
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32.6809	1.03919
$990$	0	0
$991$	−0.214286	−0.00680703	−0.00340351	−	0.999994i	$-0.501083\pi$
$991$	−0.214286	−0.00680703	−0.00340351	+	0.999994i	$0.501083\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	34.3147	1.08785
$996$	0	0
$997$	19.3093	0.611532	0.305766	−	0.952107i	$-0.401087\pi$
$997$	19.3093	0.611532	0.305766	+	0.952107i	$0.401087\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.t.1.4	yes	6
3.2	odd	2	inner	4788.2.a.t.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.t.1.3	✓	6	3.2	odd	2	inner
4788.2.a.t.1.4	yes	6	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)

\(f(q)\)	\(=\)	\(q+1.79367 q^{5} -1.00000 q^{7} +3.83284 q^{11} -3.43742 q^{13} +0.293558 q^{17} -1.00000 q^{19} +6.83306 q^{23} -1.78276 q^{25} +5.62651 q^{29} -4.78276 q^{31} -1.79367 q^{35} +11.5655 q^{37} -5.57846 q^{41} +4.78276 q^{43} -12.7051 q^{47} +1.00000 q^{49} +4.12640 q^{53} +6.87484 q^{55} +11.5655 q^{61} -6.16558 q^{65} +10.2202 q^{67} +3.53929 q^{71} -10.4404 q^{73} -3.83284 q^{77} -2.22018 q^{79} +17.7925 q^{83} +0.526544 q^{85} +5.08744 q^{89} +3.43742 q^{91} -1.79367 q^{95} +9.52949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} + 4 q^{13} - 6 q^{19} + 14 q^{25} - 4 q^{31} + 20 q^{37} + 4 q^{43} + 6 q^{49} - 8 q^{55} + 20 q^{61} + 12 q^{67} + 36 q^{73} + 36 q^{79} + 28 q^{85} - 4 q^{91} + 8 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 + 4 * q^13 - 6 * q^19 + 14 * q^25 - 4 * q^31 + 20 * q^37 + 4 * q^43 + 6 * q^49 - 8 * q^55 + 20 * q^61 + 12 * q^67 + 36 * q^73 + 36 * q^79 + 28 * q^85 - 4 * q^91 + 8 * q^97

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