LMFDB - Embedded newform 4900.2.e.t.2549.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(2549,4900)] 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("4900.2549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-16,0,8,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0, 0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 700)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.3
Root			$-2.19869i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.t.2549.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.364448i q^{3} +2.86718 q^{9} +1.36445 q^{11} -2.63555i q^{13} -2.23163i q^{17} +6.23163 q^{19} +6.59607i q^{23} -2.13828i q^{27} -5.50273 q^{29} +4.50273 q^{31} -0.497270i q^{33} +2.09334i q^{37} -0.960522 q^{39} +9.32497 q^{41} -1.86718i q^{43} +6.86718i q^{47} -0.813312 q^{51} -10.1383i q^{53} -2.27110i q^{57} +1.63555 q^{59} +0.0394782 q^{61} +6.72890i q^{67} +2.40393 q^{69} -6.27110 q^{71} -4.00000i q^{73} -5.32497 q^{79} +7.82224 q^{81} -14.7738i q^{83} +2.00546i q^{87} -0.867178 q^{89} -1.64101i q^{93} +16.0988i q^{97} +3.91211 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99}+O(q^{100}) $ 6 * q - 16 * q^9 + 8 * q^11 + 4 * q^19 - 6 * q^31 + 28 * q^39 + 22 * q^41 - 6 * q^51 + 10 * q^59 + 34 * q^61 + 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 + 28 * q^89 - 42 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 0.364448i	− 0.210414i	−0.994450	−	0.105207i	$-0.966449\pi$
$3$	− 0.364448i	− 0.210414i	0.994450	−	0.105207i	$-0.0335505\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.86718	0.955726
$10$	0	0
$11$	1.36445	0.411397	0.205698	−	0.978615i	$-0.434053\pi$
$11$	1.36445	0.411397	0.205698	+	0.978615i	$0.434053\pi$
$12$	0	0
$13$	− 2.63555i	− 0.730971i	−0.930817	−	0.365485i	$-0.880903\pi$
$13$	− 2.63555i	− 0.730971i	0.930817	−	0.365485i	$-0.119097\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.23163i	− 0.541249i	−0.962685	−	0.270624i	$-0.912770\pi$
$17$	− 2.23163i	− 0.541249i	0.962685	−	0.270624i	$-0.0872301\pi$
$18$	0	0
$19$	6.23163	1.42963	0.714816	−	0.699312i	$-0.246510\pi$
$19$	6.23163	1.42963	0.714816	+	0.699312i	$0.246510\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.59607i	1.37538i	0.726006	+	0.687688i	$0.241374\pi$
$23$	6.59607i	1.37538i	−0.726006	+	0.687688i	$0.758626\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.13828i	− 0.411512i
$28$	0	0
$29$	−5.50273	−1.02183	−0.510916	−	0.859631i	$-0.670694\pi$
$29$	−5.50273	−1.02183	−0.510916	+	0.859631i	$0.670694\pi$
$30$	0	0
$31$	4.50273	0.808714	0.404357	−	0.914601i	$-0.367495\pi$
$31$	4.50273	0.808714	0.404357	+	0.914601i	$0.367495\pi$
$32$	0	0
$33$	− 0.497270i	− 0.0865637i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.09334i	0.344144i	0.985084	+	0.172072i	$0.0550461\pi$
$37$	2.09334i	0.344144i	−0.985084	+	0.172072i	$0.944954\pi$
$38$	0	0
$39$	−0.960522	−0.153807
$40$	0	0
$41$	9.32497	1.45632	0.728158	−	0.685410i	$-0.240377\pi$
$41$	9.32497	1.45632	0.728158	+	0.685410i	$0.240377\pi$
$42$	0	0
$43$	− 1.86718i	− 0.284742i	−0.989813	−	0.142371i	$-0.954527\pi$
$43$	− 1.86718i	− 0.284742i	0.989813	−	0.142371i	$-0.0454726\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.86718i	1.00168i	0.865540	+	0.500840i	$0.166976\pi$
$47$	6.86718i	1.00168i	−0.865540	+	0.500840i	$0.833024\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.813312	−0.113886
$52$	0	0
$53$	− 10.1383i	− 1.39260i	−0.717751	−	0.696300i	$-0.754828\pi$
$53$	− 10.1383i	− 1.39260i	0.717751	−	0.696300i	$-0.245172\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.27110i	− 0.300815i
$58$	0	0
$59$	1.63555	0.212931	0.106465	−	0.994316i	$-0.466047\pi$
$59$	1.63555	0.212931	0.106465	+	0.994316i	$0.466047\pi$
$60$	0	0
$61$	0.0394782	0.00505467	0.00252733	−	0.999997i	$-0.499196\pi$
$61$	0.0394782	0.00505467	0.00252733	+	0.999997i	$0.499196\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.72890i	0.822065i	0.911621	+	0.411033i	$0.134832\pi$
$67$	6.72890i	0.822065i	−0.911621	+	0.411033i	$0.865168\pi$
$68$	0	0
$69$	2.40393	0.289399
$70$	0	0
$71$	−6.27110	−0.744243	−0.372122	−	0.928184i	$-0.621370\pi$
$71$	−6.27110	−0.744243	−0.372122	+	0.928184i	$0.621370\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.32497	−0.599106	−0.299553	−	0.954080i	$-0.596838\pi$
$79$	−5.32497	−0.599106	−0.299553	+	0.954080i	$0.596838\pi$
$80$	0	0
$81$	7.82224	0.869138
$82$	0	0
$83$	− 14.7738i	− 1.62164i	−0.585296	−	0.810819i	$-0.699022\pi$
$83$	− 14.7738i	− 1.62164i	0.585296	−	0.810819i	$-0.300978\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00546i	0.215008i
$88$	0	0
$89$	−0.867178	−0.0919206	−0.0459603	−	0.998943i	$-0.514635\pi$
$89$	−0.867178	−0.0919206	−0.0459603	+	0.998943i	$0.514635\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 1.64101i	− 0.170165i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.0988i	1.63459i	0.576222	+	0.817293i	$0.304526\pi$
$97$	16.0988i	1.63459i	−0.576222	+	0.817293i	$0.695474\pi$
$98$	0	0
$99$	3.91211	0.393182
$100$	0	0
$101$	−15.6949	−1.56170	−0.780849	−	0.624719i	$-0.785213\pi$
$101$	−15.6949	−1.56170	−0.780849	+	0.624719i	$0.785213\pi$
$102$	0	0
$103$	− 13.3699i	− 1.31738i	−0.752416	−	0.658688i	$-0.771112\pi$
$103$	− 13.3699i	− 1.31738i	0.752416	−	0.658688i	$-0.228888\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.40393i	− 0.522417i	−0.965282	−	0.261209i	$-0.915879\pi$
$107$	− 5.40393i	− 0.522417i	0.965282	−	0.261209i	$-0.0841211\pi$
$108$	0	0
$109$	15.8277	1.51602	0.758009	−	0.652244i	$-0.226172\pi$
$109$	15.8277	1.51602	0.758009	+	0.652244i	$0.226172\pi$
$110$	0	0
$111$	0.762915	0.0724127
$112$	0	0
$113$	18.4238i	1.73316i	0.499036	+	0.866581i	$0.333688\pi$
$113$	18.4238i	1.73316i	−0.499036	+	0.866581i	$0.666312\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 7.55660i	− 0.698608i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.13828	−0.830753
$122$	0	0
$123$	− 3.39847i	− 0.306429i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 5.32497i	− 0.472515i	−0.971691	−	0.236257i	$-0.924079\pi$
$127$	− 5.32497i	− 0.472515i	0.971691	−	0.236257i	$-0.0759208\pi$
$128$	0	0
$129$	−0.680489	−0.0599137
$130$	0	0
$131$	16.2371	1.41864	0.709320	−	0.704886i	$-0.249002\pi$
$131$	16.2371	1.41864	0.709320	+	0.704886i	$0.249002\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.40939i	− 0.376719i	−0.982100	−	0.188360i	$-0.939683\pi$
$137$	− 4.40939i	− 0.376719i	0.982100	−	0.188360i	$-0.0603170\pi$
$138$	0	0
$139$	22.3699	1.89739	0.948695	−	0.316192i	$-0.102404\pi$
$139$	22.3699	1.89739	0.948695	+	0.316192i	$0.102404\pi$
$140$	0	0
$141$	2.50273	0.210768
$142$	0	0
$143$	− 3.59607i	− 0.300719i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−22.5082	−1.84394	−0.921971	−	0.387258i	$-0.873422\pi$
$149$	−22.5082	−1.84394	−0.921971	+	0.387258i	$0.873422\pi$
$150$	0	0
$151$	13.5027	1.09884	0.549418	−	0.835547i	$-0.314849\pi$
$151$	13.5027	1.09884	0.549418	+	0.835547i	$0.314849\pi$
$152$	0	0
$153$	− 6.39847i	− 0.517285i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.5566i	1.24155i	0.783988	+	0.620776i	$0.213182\pi$
$157$	15.5566i	1.24155i	−0.783988	+	0.620776i	$0.786818\pi$
$158$	0	0
$159$	−3.69488	−0.293023
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.40393i	0.344942i	0.985015	+	0.172471i	$0.0551751\pi$
$163$	4.40393i	0.344942i	−0.985015	+	0.172471i	$0.944825\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.09880i	0.704087i	0.935984	+	0.352043i	$0.114513\pi$
$167$	9.09880i	0.704087i	−0.935984	+	0.352043i	$0.885487\pi$
$168$	0	0
$169$	6.05387	0.465682
$170$	0	0
$171$	17.8672	1.36634
$172$	0	0
$173$	− 19.9605i	− 1.51757i	−0.651341	−	0.758785i	$-0.725793\pi$
$173$	− 19.9605i	− 1.51757i	0.651341	−	0.758785i	$-0.274207\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 0.596074i	− 0.0448036i
$178$	0	0
$179$	13.1921	0.986027	0.493014	−	0.870022i	$-0.335895\pi$
$179$	13.1921	0.986027	0.493014	+	0.870022i	$0.335895\pi$
$180$	0	0
$181$	10.2316	0.760511	0.380255	−	0.924882i	$-0.375836\pi$
$181$	10.2316	0.760511	0.380255	+	0.924882i	$0.375836\pi$
$182$	0	0
$183$	− 0.0143878i	− 0.00106357i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.04494i	− 0.222668i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.0055	1.01340	0.506700	−	0.862123i	$-0.330865\pi$
$191$	14.0055	1.01340	0.506700	+	0.862123i	$0.330865\pi$
$192$	0	0
$193$	− 14.4094i	− 1.03721i	−0.855014	−	0.518605i	$-0.826451\pi$
$193$	− 14.4094i	− 1.03721i	0.855014	−	0.518605i	$-0.173549\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	11.5566i	0.823373i	0.911325	+	0.411687i	$0.135060\pi$
$197$	11.5566i	0.823373i	−0.911325	+	0.411687i	$0.864940\pi$
$198$	0	0
$199$	5.40939	0.383461	0.191731	−	0.981448i	$-0.438590\pi$
$199$	5.40939	0.383461	0.191731	+	0.981448i	$0.438590\pi$
$200$	0	0
$201$	2.45233	0.172974
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	18.9121i	1.31448i
$208$	0	0
$209$	8.50273	0.588146
$210$	0	0
$211$	−10.9660	−0.754929	−0.377465	−	0.926024i	$-0.623204\pi$
$211$	−10.9660	−0.754929	−0.377465	+	0.926024i	$0.623204\pi$
$212$	0	0
$213$	2.28549i	0.156599i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.45779	−0.0985085
$220$	0	0
$221$	−5.88157	−0.395637
$222$	0	0
$223$	− 21.5566i	− 1.44354i	−0.692135	−	0.721768i	$-0.743330\pi$
$223$	− 21.5566i	− 1.44354i	0.692135	−	0.721768i	$-0.256670\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.4238i	− 0.824595i	−0.911049	−	0.412297i	$-0.864726\pi$
$227$	− 12.4238i	− 0.824595i	0.911049	−	0.412297i	$-0.135274\pi$
$228$	0	0
$229$	8.36445	0.552738	0.276369	−	0.961052i	$-0.410869\pi$
$229$	8.36445	0.552738	0.276369	+	0.961052i	$0.410869\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0055i	0.917528i	0.888558	+	0.458764i	$0.151708\pi$
$233$	14.0055i	0.917528i	−0.888558	+	0.458764i	$0.848292\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.94067i	0.126060i
$238$	0	0
$239$	−16.9605	−1.09708	−0.548542	−	0.836123i	$-0.684817\pi$
$239$	−16.9605	−1.09708	−0.548542	+	0.836123i	$0.684817\pi$
$240$	0	0
$241$	10.3304	0.665441	0.332721	−	0.943025i	$-0.392033\pi$
$241$	10.3304	0.665441	0.332721	+	0.943025i	$0.392033\pi$
$242$	0	0
$243$	− 9.26564i	− 0.594391i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 16.4238i	− 1.04502i
$248$	0	0
$249$	−5.38429	−0.341216
$250$	0	0
$251$	19.5082	1.23135	0.615673	−	0.788002i	$-0.288884\pi$
$251$	19.5082	1.23135	0.615673	+	0.788002i	$0.288884\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.4633i	1.77549i	0.460337	+	0.887744i	$0.347729\pi$
$257$	28.4633i	1.77549i	−0.460337	+	0.887744i	$0.652271\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−15.7773	−0.976591
$262$	0	0
$263$	− 28.5621i	− 1.76121i	−0.473849	−	0.880606i	$-0.657136\pi$
$263$	− 28.5621i	− 1.76121i	0.473849	−	0.880606i	$-0.342864\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0.316041i	0.0193414i
$268$	0	0
$269$	9.19761	0.560788	0.280394	−	0.959885i	$-0.409535\pi$
$269$	9.19761	0.560788	0.280394	+	0.959885i	$0.409535\pi$
$270$	0	0
$271$	3.86172	0.234583	0.117291	−	0.993098i	$-0.462579\pi$
$271$	3.86172	0.234583	0.117291	+	0.993098i	$0.462579\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.6410i	0.639356i	0.947526	+	0.319678i	$0.103575\pi$
$277$	10.6410i	0.639356i	−0.947526	+	0.319678i	$0.896425\pi$
$278$	0	0
$279$	12.9101	0.772909
$280$	0	0
$281$	1.73436	0.103463	0.0517315	−	0.998661i	$-0.483526\pi$
$281$	1.73436	0.103463	0.0517315	+	0.998661i	$0.483526\pi$
$282$	0	0
$283$	5.91211i	0.351439i	0.984440	+	0.175719i	$0.0562251\pi$
$283$	5.91211i	0.351439i	−0.984440	+	0.175719i	$0.943775\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	12.0198	0.707050
$290$	0	0
$291$	5.86718	0.343940
$292$	0	0
$293$	− 24.3250i	− 1.42108i	−0.703657	−	0.710540i	$-0.748451\pi$
$293$	− 24.3250i	− 1.42108i	0.703657	−	0.710540i	$-0.251549\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.91757i	− 0.169295i
$298$	0	0
$299$	17.3843	1.00536
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.71997i	0.328604i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 23.5226i	− 1.34250i	−0.741229	−	0.671252i	$-0.765757\pi$
$307$	− 23.5226i	− 1.34250i	0.741229	−	0.671252i	$-0.234243\pi$
$308$	0	0
$309$	−4.87264	−0.277195
$310$	0	0
$311$	−8.81877	−0.500067	−0.250033	−	0.968237i	$-0.580442\pi$
$311$	−8.81877	−0.500067	−0.250033	+	0.968237i	$0.580442\pi$
$312$	0	0
$313$	0.0933442i	0.00527612i	0.999997	+	0.00263806i	$0.000839722\pi$
$313$	0.0933442i	0.00527612i	−0.999997	+	0.00263806i	$0.999160\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.54221i	0.198950i	0.995040	+	0.0994751i	$0.0317164\pi$
$317$	3.54221i	0.198950i	−0.995040	+	0.0994751i	$0.968284\pi$
$318$	0	0
$319$	−7.50819	−0.420378
$320$	0	0
$321$	−1.96945	−0.109924
$322$	0	0
$323$	− 13.9067i	− 0.773787i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 5.76837i	− 0.318992i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.6499	−1.08006	−0.540029	−	0.841646i	$-0.681587\pi$
$331$	−19.6499	−1.08006	−0.540029	+	0.841646i	$0.681587\pi$
$332$	0	0
$333$	6.00199i	0.328907i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 23.0449i	− 1.25534i	−0.778480	−	0.627669i	$-0.784009\pi$
$337$	− 23.0449i	− 1.25534i	0.778480	−	0.627669i	$-0.215991\pi$
$338$	0	0
$339$	6.71451	0.364682
$340$	0	0
$341$	6.14374	0.332702
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 22.5621i	− 1.21119i	−0.795771	−	0.605597i	$-0.792934\pi$
$347$	− 22.5621i	− 1.21119i	0.795771	−	0.605597i	$-0.207066\pi$
$348$	0	0
$349$	−14.3250	−0.766798	−0.383399	−	0.923583i	$-0.625247\pi$
$349$	−14.3250	−0.766798	−0.383399	+	0.923583i	$0.625247\pi$
$350$	0	0
$351$	−5.63555	−0.300804
$352$	0	0
$353$	− 32.7003i	− 1.74046i	−0.492643	−	0.870232i	$-0.663969\pi$
$353$	− 32.7003i	− 1.74046i	0.492643	−	0.870232i	$-0.336031\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.1581	1.38057	0.690287	−	0.723536i	$-0.257484\pi$
$359$	26.1581	1.38057	0.690287	+	0.723536i	$0.257484\pi$
$360$	0	0
$361$	19.8332	1.04385
$362$	0	0
$363$	3.33043i	0.174802i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 25.2316i	− 1.31708i	−0.752546	−	0.658540i	$-0.771174\pi$
$367$	− 25.2316i	− 1.31708i	0.752546	−	0.658540i	$-0.228826\pi$
$368$	0	0
$369$	26.7363	1.39184
$370$	0	0
$371$	0	0
$372$	0	0
$373$	5.69488i	0.294870i	0.989072	+	0.147435i	$0.0471017\pi$
$373$	5.69488i	0.294870i	−0.989072	+	0.147435i	$0.952898\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.5027i	0.746929i
$378$	0	0
$379$	−14.9660	−0.768751	−0.384375	−	0.923177i	$-0.625583\pi$
$379$	−14.9660	−0.768751	−0.384375	+	0.923177i	$0.625583\pi$
$380$	0	0
$381$	−1.94067	−0.0994238
$382$	0	0
$383$	24.4776i	1.25075i	0.780325	+	0.625374i	$0.215054\pi$
$383$	24.4776i	1.25075i	−0.780325	+	0.625374i	$0.784946\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.35353i	− 0.272135i
$388$	0	0
$389$	31.2460	1.58424	0.792118	−	0.610368i	$-0.208978\pi$
$389$	31.2460	1.58424	0.792118	+	0.610368i	$0.208978\pi$
$390$	0	0
$391$	14.7200	0.744421
$392$	0	0
$393$	− 5.91757i	− 0.298502i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.13282i	− 0.0568547i	−0.999596	−	0.0284274i	$-0.990950\pi$
$397$	− 1.13282i	− 0.0568547i	0.999596	−	0.0284274i	$-0.00904993\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.82224	−0.190874	−0.0954368	−	0.995435i	$-0.530425\pi$
$401$	−3.82224	−0.190874	−0.0954368	+	0.995435i	$0.530425\pi$
$402$	0	0
$403$	− 11.8672i	− 0.591146i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.85626i	0.141580i
$408$	0	0
$409$	−33.6609	−1.66442	−0.832211	−	0.554459i	$-0.812925\pi$
$409$	−33.6609	−1.66442	−0.832211	+	0.554459i	$0.812925\pi$
$410$	0	0
$411$	−1.60699	−0.0792671
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.15267i	− 0.399238i
$418$	0	0
$419$	0.497270	0.0242933	0.0121466	−	0.999926i	$-0.496134\pi$
$419$	0.497270	0.0242933	0.0121466	+	0.999926i	$0.496134\pi$
$420$	0	0
$421$	−1.28003	−0.0623850	−0.0311925	−	0.999513i	$-0.509930\pi$
$421$	−1.28003	−0.0623850	−0.0311925	+	0.999513i	$0.509930\pi$
$422$	0	0
$423$	19.6894i	0.957332i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.31058	−0.0632755
$430$	0	0
$431$	−10.2910	−0.495698	−0.247849	−	0.968799i	$-0.579724\pi$
$431$	−10.2910	−0.495698	−0.247849	+	0.968799i	$0.579724\pi$
$432$	0	0
$433$	− 29.8870i	− 1.43628i	−0.695899	−	0.718139i	$-0.744994\pi$
$433$	− 29.8870i	− 1.43628i	0.695899	−	0.718139i	$-0.255006\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.1043i	1.96628i
$438$	0	0
$439$	5.04841	0.240947	0.120474	−	0.992717i	$-0.461559\pi$
$439$	5.04841	0.240947	0.120474	+	0.992717i	$0.461559\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11.7738i	0.559392i	0.960089	+	0.279696i	$0.0902336\pi$
$443$	11.7738i	0.559392i	−0.960089	+	0.279696i	$0.909766\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.20307i	0.387992i
$448$	0	0
$449$	11.3843	0.537258	0.268629	−	0.963244i	$-0.413429\pi$
$449$	11.3843	0.537258	0.268629	+	0.963244i	$0.413429\pi$
$450$	0	0
$451$	12.7234	0.599123
$452$	0	0
$453$	− 4.92104i	− 0.231211i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	37.2515i	1.74255i	0.490795	+	0.871275i	$0.336706\pi$
$457$	37.2515i	1.74255i	−0.490795	+	0.871275i	$0.663294\pi$
$458$	0	0
$459$	−4.77184	−0.222731
$460$	0	0
$461$	−15.0449	−0.700713	−0.350356	−	0.936616i	$-0.613939\pi$
$461$	−15.0449	−0.700713	−0.350356	+	0.936616i	$0.613939\pi$
$462$	0	0
$463$	31.2031i	1.45013i	0.688681	+	0.725065i	$0.258190\pi$
$463$	31.2031i	1.45013i	−0.688681	+	0.725065i	$0.741810\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.7684i	0.868497i	0.900793	+	0.434248i	$0.142986\pi$
$467$	18.7684i	0.868497i	−0.900793	+	0.434248i	$0.857014\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.66957	0.261240
$472$	0	0
$473$	− 2.54767i	− 0.117142i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 29.0683i	− 1.33094i
$478$	0	0
$479$	0.714508	0.0326467	0.0163234	−	0.999867i	$-0.494804\pi$
$479$	0.714508	0.0326467	0.0163234	+	0.999867i	$0.494804\pi$
$480$	0	0
$481$	5.51712	0.251559
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.6949i	0.756517i	0.925700	+	0.378259i	$0.123477\pi$
$487$	16.6949i	0.756517i	−0.925700	+	0.378259i	$0.876523\pi$
$488$	0	0
$489$	1.60500	0.0725807
$490$	0	0
$491$	6.19761	0.279694	0.139847	−	0.990173i	$-0.455339\pi$
$491$	6.19761	0.279694	0.139847	+	0.990173i	$0.455339\pi$
$492$	0	0
$493$	12.2800i	0.553065i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−14.4687	−0.647708	−0.323854	−	0.946107i	$-0.604979\pi$
$499$	−14.4687	−0.647708	−0.323854	+	0.946107i	$0.604979\pi$
$500$	0	0
$501$	3.31604	0.148150
$502$	0	0
$503$	30.6949i	1.36862i	0.729193	+	0.684308i	$0.239896\pi$
$503$	30.6949i	1.36862i	−0.729193	+	0.684308i	$0.760104\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.20632i	− 0.0979861i
$508$	0	0
$509$	−44.4742	−1.97128	−0.985641	−	0.168852i	$-0.945994\pi$
$509$	−44.4742	−1.97128	−0.985641	+	0.168852i	$0.945994\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 13.3250i	− 0.588312i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.36991i	0.412088i
$518$	0	0
$519$	−7.27457	−0.319318
$520$	0	0
$521$	−12.6499	−0.554204	−0.277102	−	0.960841i	$-0.589374\pi$
$521$	−12.6499	−0.554204	−0.277102	+	0.960841i	$0.589374\pi$
$522$	0	0
$523$	− 10.2711i	− 0.449124i	−0.974460	−	0.224562i	$-0.927905\pi$
$523$	− 10.2711i	− 0.449124i	0.974460	−	0.224562i	$-0.0720951\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 10.0484i	− 0.437715i
$528$	0	0
$529$	−20.5082	−0.891660
$530$	0	0
$531$	4.68942	0.203503
$532$	0	0
$533$	− 24.5764i	− 1.06452i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 4.80785i	− 0.207474i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.13828	−0.0919319	−0.0459660	−	0.998943i	$-0.514637\pi$
$541$	−2.13828	−0.0919319	−0.0459660	+	0.998943i	$0.514637\pi$
$542$	0	0
$543$	− 3.72890i	− 0.160022i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.0198i	1.36907i	0.728980	+	0.684535i	$0.239995\pi$
$547$	32.0198i	1.36907i	−0.728980	+	0.684535i	$0.760005\pi$
$548$	0	0
$549$	0.113191	0.00483088
$550$	0	0
$551$	−34.2910	−1.46084
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.33043i	0.352972i	0.984303	+	0.176486i	$0.0564730\pi$
$557$	8.33043i	0.352972i	−0.984303	+	0.176486i	$0.943527\pi$
$558$	0	0
$559$	−4.92104	−0.208138
$560$	0	0
$561$	−1.10972	−0.0468525
$562$	0	0
$563$	− 19.4633i	− 0.820278i	−0.912023	−	0.410139i	$-0.865480\pi$
$563$	− 19.4633i	− 0.820278i	0.912023	−	0.410139i	$-0.134520\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.8925	−0.792014	−0.396007	−	0.918247i	$-0.629604\pi$
$569$	−18.8925	−0.792014	−0.396007	+	0.918247i	$0.629604\pi$
$570$	0	0
$571$	−18.6301	−0.779645	−0.389823	−	0.920890i	$-0.627464\pi$
$571$	−18.6301	−0.779645	−0.389823	+	0.920890i	$0.627464\pi$
$572$	0	0
$573$	− 5.10426i	− 0.213234i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.5926i	0.774020i	0.922075	+	0.387010i	$0.126492\pi$
$577$	18.5926i	0.774020i	−0.922075	+	0.387010i	$0.873508\pi$
$578$	0	0
$579$	−5.25147	−0.218244
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 13.8332i	− 0.572911i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 10.1383i	− 0.418452i	−0.977867	−	0.209226i	$-0.932906\pi$
$587$	− 10.1383i	− 0.418452i	0.977867	−	0.209226i	$-0.0670944\pi$
$588$	0	0
$589$	28.0593	1.15616
$590$	0	0
$591$	4.21178	0.173249
$592$	0	0
$593$	− 17.0055i	− 0.698331i	−0.937061	−	0.349165i	$-0.886465\pi$
$593$	− 17.0055i	− 0.698331i	0.937061	−	0.349165i	$-0.113535\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.97144i	− 0.0806857i
$598$	0	0
$599$	8.18669	0.334499	0.167250	−	0.985915i	$-0.446511\pi$
$599$	8.18669	0.334499	0.167250	+	0.985915i	$0.446511\pi$
$600$	0	0
$601$	−35.3359	−1.44138	−0.720690	−	0.693257i	$-0.756175\pi$
$601$	−35.3359	−1.44138	−0.720690	+	0.693257i	$0.756175\pi$
$602$	0	0
$603$	19.2929i	0.785669i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 28.2316i	− 1.14589i	−0.819595	−	0.572943i	$-0.805802\pi$
$607$	− 28.2316i	− 1.14589i	0.819595	−	0.572943i	$-0.194198\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.0988	0.732199
$612$	0	0
$613$	31.1581i	1.25846i	0.777217	+	0.629232i	$0.216631\pi$
$613$	31.1581i	1.25846i	−0.777217	+	0.629232i	$0.783369\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.1527i	1.57623i	0.615530	+	0.788114i	$0.288942\pi$
$617$	39.1527i	1.57623i	−0.615530	+	0.788114i	$0.711058\pi$
$618$	0	0
$619$	39.3843	1.58299	0.791494	−	0.611177i	$-0.209304\pi$
$619$	39.3843	1.58299	0.791494	+	0.611177i	$0.209304\pi$
$620$	0	0
$621$	14.1043	0.565985
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.09880i	− 0.123754i
$628$	0	0
$629$	4.67156	0.186267
$630$	0	0
$631$	22.7003	0.903686	0.451843	−	0.892097i	$-0.350767\pi$
$631$	22.7003	0.903686	0.451843	+	0.892097i	$0.350767\pi$
$632$	0	0
$633$	3.99653i	0.158848i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−17.9804	−0.711292
$640$	0	0
$641$	8.41285	0.332288	0.166144	−	0.986102i	$-0.446868\pi$
$641$	8.41285	0.332288	0.166144	+	0.986102i	$0.446868\pi$
$642$	0	0
$643$	29.7398i	1.17282i	0.810013	+	0.586412i	$0.199460\pi$
$643$	29.7398i	1.17282i	−0.810013	+	0.586412i	$0.800540\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 21.6859i	− 0.852563i	−0.904591	−	0.426281i	$-0.859823\pi$
$647$	− 21.6859i	− 0.852563i	0.904591	−	0.426281i	$-0.140177\pi$
$648$	0	0
$649$	2.23163	0.0875990
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 30.0702i	− 1.17674i	−0.808592	−	0.588370i	$-0.799770\pi$
$653$	− 30.0702i	− 1.17674i	0.808592	−	0.588370i	$-0.200230\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 11.4687i	− 0.447437i
$658$	0	0
$659$	−2.37537	−0.0925311	−0.0462656	−	0.998929i	$-0.514732\pi$
$659$	−2.37537	−0.0925311	−0.0462656	+	0.998929i	$0.514732\pi$
$660$	0	0
$661$	−14.4633	−0.562555	−0.281278	−	0.959626i	$-0.590758\pi$
$661$	−14.4633	−0.562555	−0.281278	+	0.959626i	$0.590758\pi$
$662$	0	0
$663$	2.14353i	0.0832476i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 36.2964i	− 1.40540i
$668$	0	0
$669$	−7.85626	−0.303741
$670$	0	0
$671$	0.0538660	0.00207947
$672$	0	0
$673$	− 8.99653i	− 0.346791i	−0.984852	−	0.173395i	$-0.944526\pi$
$673$	− 8.99653i	− 0.346791i	0.984852	−	0.173395i	$-0.0554738\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.49181i	0.287934i	0.989583	+	0.143967i	$0.0459859\pi$
$677$	7.49181i	0.287934i	−0.989583	+	0.143967i	$0.954014\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−4.52782	−0.173506
$682$	0	0
$683$	42.3250i	1.61952i	0.586761	+	0.809760i	$0.300403\pi$
$683$	42.3250i	1.61952i	−0.586761	+	0.809760i	$0.699597\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.04841i	− 0.116304i
$688$	0	0
$689$	−26.7200	−1.01795
$690$	0	0
$691$	28.6465	1.08976	0.544882	−	0.838513i	$-0.316575\pi$
$691$	28.6465	1.08976	0.544882	+	0.838513i	$0.316575\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 20.8098i	− 0.788229i
$698$	0	0
$699$	5.10426	0.193061
$700$	0	0
$701$	−20.4292	−0.771601	−0.385801	−	0.922582i	$-0.626075\pi$
$701$	−20.4292	−0.771601	−0.385801	+	0.922582i	$0.626075\pi$
$702$	0	0
$703$	13.0449i	0.491999i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−28.8277	−1.08265	−0.541323	−	0.840814i	$-0.682077\pi$
$709$	−28.8277	−1.08265	−0.541323	+	0.840814i	$0.682077\pi$
$710$	0	0
$711$	−15.2676	−0.572581
$712$	0	0
$713$	29.7003i	1.11229i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.18123i	0.230842i
$718$	0	0
$719$	−31.3106	−1.16769	−0.583844	−	0.811866i	$-0.698452\pi$
$719$	−31.3106	−1.16769	−0.583844	+	0.811866i	$0.698452\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 3.76490i	− 0.140018i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.17230i	− 0.303094i	−0.988450	−	0.151547i	$-0.951575\pi$
$727$	− 8.17230i	− 0.303094i	0.988450	−	0.151547i	$-0.0484255\pi$
$728$	0	0
$729$	20.0899	0.744069
$730$	0	0
$731$	−4.16684	−0.154116
$732$	0	0
$733$	21.4183i	0.791103i	0.918444	+	0.395552i	$0.129447\pi$
$733$	21.4183i	0.791103i	−0.918444	+	0.395552i	$0.870553\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.18123i	0.338195i
$738$	0	0
$739$	−4.35006	−0.160020	−0.0800098	−	0.996794i	$-0.525495\pi$
$739$	−4.35006	−0.160020	−0.0800098	+	0.996794i	$0.525495\pi$
$740$	0	0
$741$	−5.98561	−0.219887
$742$	0	0
$743$	− 6.53328i	− 0.239683i	−0.992793	−	0.119841i	$-0.961761\pi$
$743$	− 6.53328i	− 0.239683i	0.992793	−	0.119841i	$-0.0382386\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 42.3592i	− 1.54984i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	34.4831	1.25831	0.629153	−	0.777281i	$-0.283402\pi$
$751$	34.4831	1.25831	0.629153	+	0.777281i	$0.283402\pi$
$752$	0	0
$753$	− 7.10972i	− 0.259093i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 44.2425i	− 1.60802i	−0.594614	−	0.804011i	$-0.702695\pi$
$757$	− 44.2425i	− 1.60802i	0.594614	−	0.804011i	$-0.297305\pi$
$758$	0	0
$759$	3.28003	0.119058
$760$	0	0
$761$	−23.8188	−0.863430	−0.431715	−	0.902010i	$-0.642091\pi$
$761$	−23.8188	−0.863430	−0.431715	+	0.902010i	$0.642091\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 4.31058i	− 0.155646i
$768$	0	0
$769$	0.158128	0.00570225	0.00285113	−	0.999996i	$-0.499092\pi$
$769$	0.158128	0.00570225	0.00285113	+	0.999996i	$0.499092\pi$
$770$	0	0
$771$	10.3734	0.373588
$772$	0	0
$773$	17.7991i	0.640191i	0.947385	+	0.320095i	$0.103715\pi$
$773$	17.7991i	0.640191i	−0.947385	+	0.320095i	$0.896285\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	58.1097	2.08200
$780$	0	0
$781$	−8.55660	−0.306179
$782$	0	0
$783$	11.7664i	0.420496i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 41.5226i	− 1.48012i	−0.672541	−	0.740060i	$-0.734797\pi$
$787$	− 41.5226i	− 1.48012i	0.672541	−	0.740060i	$-0.265203\pi$
$788$	0	0
$789$	−10.4094	−0.370584
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 0.104047i	− 0.00369481i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.0933i	0.888852i	0.895816	+	0.444426i	$0.146592\pi$
$797$	25.0933i	0.888852i	−0.895816	+	0.444426i	$0.853408\pi$
$798$	0	0
$799$	15.3250	0.542158
$800$	0	0
$801$	−2.48635	−0.0878509
$802$	0	0
$803$	− 5.45779i	− 0.192601i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.35205i	− 0.117998i
$808$	0	0
$809$	−12.6949	−0.446328	−0.223164	−	0.974781i	$-0.571639\pi$
$809$	−12.6949	−0.446328	−0.223164	+	0.974781i	$0.571639\pi$
$810$	0	0
$811$	17.9749	0.631184	0.315592	−	0.948895i	$-0.397797\pi$
$811$	17.9749	0.631184	0.315592	+	0.948895i	$0.397797\pi$
$812$	0	0
$813$	− 1.40740i	− 0.0493595i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 11.6356i	− 0.407076i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	20.0683	0.700387	0.350193	−	0.936677i	$-0.386116\pi$
$821$	20.0683	0.700387	0.350193	+	0.936677i	$0.386116\pi$
$822$	0	0
$823$	10.6105i	0.369857i	0.982752	+	0.184929i	$0.0592054\pi$
$823$	10.6105i	0.369857i	−0.982752	+	0.184929i	$0.940795\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34.7793i	1.20939i	0.796455	+	0.604697i	$0.206706\pi$
$827$	34.7793i	1.20939i	−0.796455	+	0.604697i	$0.793294\pi$
$828$	0	0
$829$	26.2282	0.910942	0.455471	−	0.890251i	$-0.349471\pi$
$829$	26.2282	0.910942	0.455471	+	0.890251i	$0.349471\pi$
$830$	0	0
$831$	3.87810	0.134530
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 9.62810i	− 0.332796i
$838$	0	0
$839$	−33.9210	−1.17108	−0.585542	−	0.810642i	$-0.699118\pi$
$839$	−33.9210	−1.17108	−0.585542	+	0.810642i	$0.699118\pi$
$840$	0	0
$841$	1.28003	0.0441391
$842$	0	0
$843$	− 0.632082i	− 0.0217701i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.15466	0.0739477
$850$	0	0
$851$	−13.8079	−0.473327
$852$	0	0
$853$	57.6698i	1.97458i	0.158942	+	0.987288i	$0.449192\pi$
$853$	57.6698i	1.97458i	−0.158942	+	0.987288i	$0.550808\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.4039i	1.20938i	0.796463	+	0.604688i	$0.206702\pi$
$857$	35.4039i	1.20938i	−0.796463	+	0.604688i	$0.793298\pi$
$858$	0	0
$859$	13.4434	0.458683	0.229342	−	0.973346i	$-0.426343\pi$
$859$	13.4434	0.458683	0.229342	+	0.973346i	$0.426343\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.06478i	0.240488i	0.992744	+	0.120244i	$0.0383677\pi$
$863$	7.06478i	0.240488i	−0.992744	+	0.120244i	$0.961632\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.38061i	− 0.148773i
$868$	0	0
$869$	−7.26564	−0.246470
$870$	0	0
$871$	17.7344	0.600906
$872$	0	0
$873$	46.1581i	1.56222i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.0055i	0.506698i	0.967375	+	0.253349i	$0.0815322\pi$
$877$	15.0055i	0.506698i	−0.967375	+	0.253349i	$0.918468\pi$
$878$	0	0
$879$	−8.86519	−0.299015
$880$	0	0
$881$	−39.3699	−1.32641	−0.663203	−	0.748440i	$-0.730803\pi$
$881$	−39.3699	−1.32641	−0.663203	+	0.748440i	$0.730803\pi$
$882$	0	0
$883$	− 58.2229i	− 1.95936i	−0.200574	−	0.979679i	$-0.564281\pi$
$883$	− 58.2229i	− 1.95936i	0.200574	−	0.979679i	$-0.435719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.2460i	0.847678i	0.905738	+	0.423839i	$0.139318\pi$
$887$	25.2460i	0.847678i	−0.905738	+	0.423839i	$0.860682\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.6730	0.357560
$892$	0	0
$893$	42.7937i	1.43204i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.33567i	− 0.211542i
$898$	0	0
$899$	−24.7773	−0.826369
$900$	0	0
$901$	−22.6248	−0.753743
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	27.1867i	0.902719i	0.892342	+	0.451360i	$0.149061\pi$
$907$	27.1867i	0.902719i	−0.892342	+	0.451360i	$0.850939\pi$
$908$	0	0
$909$	−45.0000	−1.49256
$910$	0	0
$911$	−40.7882	−1.35137	−0.675687	−	0.737189i	$-0.736153\pi$
$911$	−40.7882	−1.35137	−0.675687	+	0.737189i	$0.736153\pi$
$912$	0	0
$913$	− 20.1581i	− 0.667137i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	47.0737	1.55282	0.776409	−	0.630229i	$-0.217039\pi$
$919$	47.0737	1.55282	0.776409	+	0.630229i	$0.217039\pi$
$920$	0	0
$921$	−8.57276	−0.282482
$922$	0	0
$923$	16.5278i	0.544020i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 38.3339i	− 1.25905i
$928$	0	0
$929$	−17.9911	−0.590268	−0.295134	−	0.955456i	$-0.595364\pi$
$929$	−17.9911	−0.590268	−0.295134	+	0.955456i	$0.595364\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.21398i	0.105221i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 37.7003i	− 1.23162i	−0.787896	−	0.615808i	$-0.788830\pi$
$937$	− 37.7003i	− 1.23162i	0.787896	−	0.615808i	$-0.211170\pi$
$938$	0	0
$939$	0.0340191	0.00111017
$940$	0	0
$941$	4.83316	0.157556	0.0787782	−	0.996892i	$-0.474898\pi$
$941$	4.83316	0.157556	0.0787782	+	0.996892i	$0.474898\pi$
$942$	0	0
$943$	61.5082i	2.00298i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.83663i	0.0921780i	0.998937	+	0.0460890i	$0.0146758\pi$
$947$	2.83663i	0.0921780i	−0.998937	+	0.0460890i	$0.985324\pi$
$948$	0	0
$949$	−10.5422	−0.342215
$950$	0	0
$951$	1.29095	0.0418619
$952$	0	0
$953$	17.7991i	0.576571i	0.957545	+	0.288285i	$0.0930852\pi$
$953$	17.7991i	0.576571i	−0.957545	+	0.288285i	$0.906915\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.73634i	0.0884535i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−10.7254	−0.345982
$962$	0	0
$963$	− 15.4940i	− 0.499288i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	15.1581i	0.487453i	0.969844	+	0.243726i	$0.0783698\pi$
$967$	15.1581i	0.487453i	−0.969844	+	0.243726i	$0.921630\pi$
$968$	0	0
$969$	−5.06825	−0.162816
$970$	0	0
$971$	−47.3843	−1.52063	−0.760317	−	0.649552i	$-0.774956\pi$
$971$	−47.3843	−1.52063	−0.760317	+	0.649552i	$0.774956\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 23.7200i	− 0.758869i	−0.925219	−	0.379434i	$-0.876119\pi$
$977$	− 23.7200i	− 0.758869i	0.925219	−	0.379434i	$-0.123881\pi$
$978$	0	0
$979$	−1.18322	−0.0378158
$980$	0	0
$981$	45.3808	1.44890
$982$	0	0
$983$	34.4543i	1.09892i	0.835519	+	0.549461i	$0.185167\pi$
$983$	34.4543i	1.09892i	−0.835519	+	0.549461i	$0.814833\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.3160	0.391627
$990$	0	0
$991$	−41.6609	−1.32340	−0.661700	−	0.749768i	$-0.730165\pi$
$991$	−41.6609	−1.32340	−0.661700	+	0.749768i	$0.730165\pi$
$992$	0	0
$993$	7.16138i	0.227260i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 24.7882i	− 0.785051i	−0.919741	−	0.392525i	$-0.871601\pi$
$997$	− 24.7882i	− 0.785051i	0.919741	−	0.392525i	$-0.128399\pi$
$998$	0	0
$999$	4.47616	0.141619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.t.2549.3		6
5.2	odd	4		4900.2.a.ba.1.2		3
5.3	odd	4		4900.2.a.bc.1.2		3
5.4	even	2	inner	4900.2.e.t.2549.4		6
7.2	even	3		700.2.r.d.249.4		12
7.4	even	3		700.2.r.d.149.3		12
7.6	odd	2		4900.2.e.s.2549.4		6
35.2	odd	12		700.2.i.e.501.2	yes	6
35.4	even	6		700.2.r.d.149.4		12
35.9	even	6		700.2.r.d.249.3		12
35.13	even	4		4900.2.a.bb.1.2		3
35.18	odd	12		700.2.i.d.401.2	✓	6
35.23	odd	12		700.2.i.d.501.2	yes	6
35.27	even	4		4900.2.a.bd.1.2		3
35.32	odd	12		700.2.i.e.401.2	yes	6
35.34	odd	2		4900.2.e.s.2549.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.2	✓	6	35.18	odd	12
700.2.i.d.501.2	yes	6	35.23	odd	12
700.2.i.e.401.2	yes	6	35.32	odd	12
700.2.i.e.501.2	yes	6	35.2	odd	12
700.2.r.d.149.3		12	7.4	even	3
700.2.r.d.149.4		12	35.4	even	6
700.2.r.d.249.3		12	35.9	even	6
700.2.r.d.249.4		12	7.2	even	3
4900.2.a.ba.1.2		3	5.2	odd	4
4900.2.a.bb.1.2		3	35.13	even	4
4900.2.a.bc.1.2		3	5.3	odd	4
4900.2.a.bd.1.2		3	35.27	even	4
4900.2.e.s.2549.3		6	35.34	odd	2
4900.2.e.s.2549.4		6	7.6	odd	2
4900.2.e.t.2549.3		6	1.1	even	1	trivial
4900.2.e.t.2549.4		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.364448i q^{3} +2.86718 q^{9} +1.36445 q^{11} -2.63555i q^{13} -2.23163i q^{17} +6.23163 q^{19} +6.59607i q^{23} -2.13828i q^{27} -5.50273 q^{29} +4.50273 q^{31} -0.497270i q^{33} +2.09334i q^{37} -0.960522 q^{39} +9.32497 q^{41} -1.86718i q^{43} +6.86718i q^{47} -0.813312 q^{51} -10.1383i q^{53} -2.27110i q^{57} +1.63555 q^{59} +0.0394782 q^{61} +6.72890i q^{67} +2.40393 q^{69} -6.27110 q^{71} -4.00000i q^{73} -5.32497 q^{79} +7.82224 q^{81} -14.7738i q^{83} +2.00546i q^{87} -0.867178 q^{89} -1.64101i q^{93} +16.0988i q^{97} +3.91211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{9} + 8 q^{11} + 4 q^{19} - 6 q^{31} + 28 q^{39} + 22 q^{41} - 6 q^{51} + 10 q^{59} + 34 q^{61} + 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} + 28 q^{89} - 42 q^{99}+O(q^{100}) \) 6 * q - 16 * q^9 + 8 * q^11 + 4 * q^19 - 6 * q^31 + 28 * q^39 + 22 * q^41 - 6 * q^51 + 10 * q^59 + 34 * q^61 + 48 * q^69 - 38 * q^71 + 2 * q^79 + 46 * q^81 + 28 * q^89 - 42 * q^99

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