LMFDB - Embedded newform 4212.2.a.j.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4212,2,Mod(1,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("4212.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4212 = 2^{2} \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4212.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,7,0,-2,0,0,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.6329893314$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.8655345.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 10x^{3} + 11x^{2} + 24x - 9 $ x^5 - 2x^4 - 10x^3 + 11x^2 + 24x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 468)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.45396$ of defining polynomial
Character	$\chi$	$=$	4212.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.45396 q^{5} -4.47585 q^{7} -4.34442 q^{11} +1.00000 q^{13} +2.89047 q^{17} +5.31660 q^{19} +0.599782 q^{23} +14.8377 q^{25} -0.635109 q^{29} +2.63511 q^{31} -19.9353 q^{35} -1.47585 q^{37} -5.93828 q^{41} -3.51215 q^{43} +3.85417 q^{47} +13.0333 q^{49} +1.32698 q^{53} -19.3499 q^{55} +3.38072 q^{59} -0.512149 q^{61} +4.45396 q^{65} -5.59881 q^{67} +11.9079 q^{71} +5.96371 q^{73} +19.4450 q^{77} +15.1906 q^{79} +7.56349 q^{83} +12.8740 q^{85} +6.88454 q^{89} -4.47585 q^{91} +23.6799 q^{95} +10.7511 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 7 q^{5} - 2 q^{7} - 5 q^{11} + 5 q^{13} + 13 q^{17} + q^{19} + 8 q^{25} + 12 q^{29} - 2 q^{31} - 5 q^{35} + 13 q^{37} + 4 q^{41} - 8 q^{43} + 7 q^{47} + 15 q^{49} + 19 q^{53} - 21 q^{55} + 11 q^{59}+ \cdots + 4 q^{97}+O(q^{100}) $ 5 * q + 7 * q^5 - 2 * q^7 - 5 * q^11 + 5 * q^13 + 13 * q^17 + q^19 + 8 * q^25 + 12 * q^29 - 2 * q^31 - 5 * q^35 + 13 * q^37 + 4 * q^41 - 8 * q^43 + 7 * q^47 + 15 * q^49 + 19 * q^53 - 21 * q^55 + 11 * q^59 + 7 * q^61 + 7 * q^65 - 2 * q^67 + 29 * q^71 + 19 * q^73 + 7 * q^77 - 5 * q^79 + 24 * q^83 + 9 * q^85 + 19 * q^89 - 2 * q^91 + 10 * q^95 + 4 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	4.45396	1.99187	0.995935	−	0.0900774i	$-0.0287114\pi$
$5$	4.45396	1.99187	0.995935	+	0.0900774i	$0.0287114\pi$
$6$	0	0
$7$	−4.47585	−1.69171	−0.845857	−	0.533410i	$-0.820910\pi$
$7$	−4.47585	−1.69171	−0.845857	+	0.533410i	$0.820910\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.34442	−1.30989	−0.654946	−	0.755675i	$-0.727309\pi$
$11$	−4.34442	−1.30989	−0.654946	+	0.755675i	$0.727309\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.89047	0.701041	0.350521	−	0.936555i	$-0.386005\pi$
$17$	2.89047	0.701041	0.350521	+	0.936555i	$0.386005\pi$
$18$	0	0
$19$	5.31660	1.21971	0.609856	−	0.792512i	$-0.291227\pi$
$19$	5.31660	1.21971	0.609856	+	0.792512i	$0.291227\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.599782	0.125063	0.0625316	−	0.998043i	$-0.480083\pi$
$23$	0.599782	0.125063	0.0625316	+	0.998043i	$0.480083\pi$
$24$	0	0
$25$	14.8377	2.96754
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.635109	−0.117937	−0.0589684	−	0.998260i	$-0.518781\pi$
$29$	−0.635109	−0.117937	−0.0589684	+	0.998260i	$0.518781\pi$
$30$	0	0
$31$	2.63511	0.473280	0.236640	−	0.971597i	$-0.423954\pi$
$31$	2.63511	0.473280	0.236640	+	0.971597i	$0.423954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−19.9353	−3.36967
$36$	0	0
$37$	−1.47585	−0.242629	−0.121314	−	0.992614i	$-0.538711\pi$
$37$	−1.47585	−0.242629	−0.121314	+	0.992614i	$0.538711\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.93828	−0.927404	−0.463702	−	0.885991i	$-0.653479\pi$
$41$	−5.93828	−0.927404	−0.463702	+	0.885991i	$0.653479\pi$
$42$	0	0
$43$	−3.51215	−0.535598	−0.267799	−	0.963475i	$-0.586296\pi$
$43$	−3.51215	−0.535598	−0.267799	+	0.963475i	$0.586296\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.85417	0.562189	0.281094	−	0.959680i	$-0.409303\pi$
$47$	3.85417	0.562189	0.281094	+	0.959680i	$0.409303\pi$
$48$	0	0
$49$	13.0333	1.86190
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.32698	0.182275	0.0911374	−	0.995838i	$-0.470950\pi$
$53$	1.32698	0.182275	0.0911374	+	0.995838i	$0.470950\pi$
$54$	0	0
$55$	−19.3499	−2.60914
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.38072	0.440132	0.220066	−	0.975485i	$-0.429373\pi$
$59$	3.38072	0.440132	0.220066	+	0.975485i	$0.429373\pi$
$60$	0	0
$61$	−0.512149	−0.0655739	−0.0327870	−	0.999462i	$-0.510438\pi$
$61$	−0.512149	−0.0655739	−0.0327870	+	0.999462i	$0.510438\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.45396	0.552445
$66$	0	0
$67$	−5.59881	−0.684004	−0.342002	−	0.939699i	$-0.611105\pi$
$67$	−5.59881	−0.684004	−0.342002	+	0.939699i	$0.611105\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.9079	1.41321	0.706605	−	0.707609i	$-0.250226\pi$
$71$	11.9079	1.41321	0.706605	+	0.707609i	$0.250226\pi$
$72$	0	0
$73$	5.96371	0.697999	0.348999	−	0.937123i	$-0.386521\pi$
$73$	5.96371	0.697999	0.348999	+	0.937123i	$0.386521\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	19.4450	2.21596
$78$	0	0
$79$	15.1906	1.70908	0.854539	−	0.519387i	$-0.173840\pi$
$79$	15.1906	1.70908	0.854539	+	0.519387i	$0.173840\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.56349	0.830201	0.415100	−	0.909776i	$-0.363746\pi$
$83$	7.56349	0.830201	0.415100	+	0.909776i	$0.363746\pi$
$84$	0	0
$85$	12.8740	1.39638
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.88454	0.729760	0.364880	−	0.931055i	$-0.381110\pi$
$89$	6.88454	0.729760	0.364880	+	0.931055i	$0.381110\pi$
$90$	0	0
$91$	−4.47585	−0.469197
$92$	0	0
$93$	0	0
$94$	0	0
$95$	23.6799	2.42951
$96$	0	0
$97$	10.7511	1.09160	0.545802	−	0.837914i	$-0.316225\pi$
$97$	10.7511	1.09160	0.545802	+	0.837914i	$0.316225\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.08701	−0.904191	−0.452096	−	0.891970i	$-0.649323\pi$
$101$	−9.08701	−0.904191	−0.452096	+	0.891970i	$0.649323\pi$
$102$	0	0
$103$	2.15925	0.212758	0.106379	−	0.994326i	$-0.466074\pi$
$103$	2.15925	0.212758	0.106379	+	0.994326i	$0.466074\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.59386	0.444105	0.222052	−	0.975035i	$-0.428724\pi$
$107$	4.59386	0.444105	0.222052	+	0.975035i	$0.428724\pi$
$108$	0	0
$109$	−12.1663	−1.16532	−0.582661	−	0.812715i	$-0.697988\pi$
$109$	−12.1663	−1.16532	−0.582661	+	0.812715i	$0.697988\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.67140	0.533521	0.266760	−	0.963763i	$-0.414047\pi$
$113$	5.67140	0.533521	0.266760	+	0.963763i	$0.414047\pi$
$114$	0	0
$115$	2.67140	0.249110
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.9373	−1.18596
$120$	0	0
$121$	7.87402	0.715820
$122$	0	0
$123$	0	0
$124$	0	0
$125$	43.8168	3.91909
$126$	0	0
$127$	5.54155	0.491733	0.245867	−	0.969304i	$-0.420928\pi$
$127$	5.54155	0.491733	0.245867	+	0.969304i	$0.420928\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.7162	−1.46050	−0.730250	−	0.683180i	$-0.760597\pi$
$131$	−16.7162	−1.46050	−0.730250	+	0.683180i	$0.760597\pi$
$132$	0	0
$133$	−23.7963	−2.06340
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0528	1.54235	0.771176	−	0.636622i	$-0.219669\pi$
$137$	18.0528	1.54235	0.771176	+	0.636622i	$0.219669\pi$
$138$	0	0
$139$	2.07769	0.176228	0.0881138	−	0.996110i	$-0.471916\pi$
$139$	2.07769	0.176228	0.0881138	+	0.996110i	$0.471916\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.34442	−0.363299
$144$	0	0
$145$	−2.82875	−0.234915
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.1573	1.24174	0.620869	−	0.783914i	$-0.286780\pi$
$149$	15.1573	1.24174	0.620869	+	0.783914i	$0.286780\pi$
$150$	0	0
$151$	−14.6785	−1.19452	−0.597258	−	0.802049i	$-0.703743\pi$
$151$	−14.6785	−1.19452	−0.597258	+	0.802049i	$0.703743\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.7367	0.942711
$156$	0	0
$157$	−8.86504	−0.707507	−0.353754	−	0.935339i	$-0.615095\pi$
$157$	−8.86504	−0.707507	−0.353754	+	0.935339i	$0.615095\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.68454	−0.211571
$162$	0	0
$163$	−0.273242	−0.0214020	−0.0107010	−	0.999943i	$-0.503406\pi$
$163$	−0.273242	−0.0214020	−0.0107010	+	0.999943i	$0.503406\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.267316	0.0206856	0.0103428	−	0.999947i	$-0.496708\pi$
$167$	0.267316	0.0206856	0.0103428	+	0.999947i	$0.496708\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.486381	0.0369788	0.0184894	−	0.999829i	$-0.494114\pi$
$173$	0.486381	0.0369788	0.0184894	+	0.999829i	$0.494114\pi$
$174$	0	0
$175$	−66.4115	−5.02024
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.4579	−0.781662	−0.390831	−	0.920462i	$-0.627812\pi$
$179$	−10.4579	−0.781662	−0.390831	+	0.920462i	$0.627812\pi$
$180$	0	0
$181$	17.8620	1.32767	0.663837	−	0.747878i	$-0.268927\pi$
$181$	17.8620	1.32767	0.663837	+	0.747878i	$0.268927\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.57339	−0.483285
$186$	0	0
$187$	−12.5574	−0.918289
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.98546	0.722522	0.361261	−	0.932465i	$-0.382346\pi$
$191$	9.98546	0.722522	0.361261	+	0.932465i	$0.382346\pi$
$192$	0	0
$193$	−3.00897	−0.216591	−0.108295	−	0.994119i	$-0.534539\pi$
$193$	−3.00897	−0.216591	−0.108295	+	0.994119i	$0.534539\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.6271	1.04214	0.521070	−	0.853514i	$-0.325533\pi$
$197$	14.6271	1.04214	0.521070	+	0.853514i	$0.325533\pi$
$198$	0	0
$199$	−15.8014	−1.12013	−0.560067	−	0.828448i	$-0.689225\pi$
$199$	−15.8014	−1.12013	−0.560067	+	0.828448i	$0.689225\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.84266	0.199515
$204$	0	0
$205$	−26.4488	−1.84727
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−23.0976	−1.59769
$210$	0	0
$211$	−21.7256	−1.49566	−0.747828	−	0.663893i	$-0.768903\pi$
$211$	−21.7256	−1.49566	−0.747828	+	0.663893i	$0.768903\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−15.6430	−1.06684
$216$	0	0
$217$	−11.7944	−0.800654
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.89047	0.194434
$222$	0	0
$223$	17.7704	1.18999	0.594996	−	0.803729i	$-0.297154\pi$
$223$	17.7704	1.18999	0.594996	+	0.803729i	$0.297154\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.1880	1.00807	0.504033	−	0.863685i	$-0.331849\pi$
$227$	15.1880	1.00807	0.504033	+	0.863685i	$0.331849\pi$
$228$	0	0
$229$	26.0403	1.72079	0.860397	−	0.509625i	$-0.170216\pi$
$229$	26.0403	1.72079	0.860397	+	0.509625i	$0.170216\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.2563	−1.06499	−0.532493	−	0.846434i	$-0.678745\pi$
$233$	−16.2563	−1.06499	−0.532493	+	0.846434i	$0.678745\pi$
$234$	0	0
$235$	17.1663	1.11981
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.0907	0.652712	0.326356	−	0.945247i	$-0.394179\pi$
$239$	10.0907	0.652712	0.326356	+	0.945247i	$0.394179\pi$
$240$	0	0
$241$	−14.2026	−0.914871	−0.457435	−	0.889243i	$-0.651232\pi$
$241$	−14.2026	−0.914871	−0.457435	+	0.889243i	$0.651232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	58.0496	3.70865
$246$	0	0
$247$	5.31660	0.338287
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.60872	0.227780	0.113890	−	0.993493i	$-0.463669\pi$
$251$	3.60872	0.227780	0.113890	+	0.993493i	$0.463669\pi$
$252$	0	0
$253$	−2.60571	−0.163819
$254$	0	0
$255$	0	0
$256$	0	0
$257$	15.5202	0.968122	0.484061	−	0.875034i	$-0.339161\pi$
$257$	15.5202	0.968122	0.484061	+	0.875034i	$0.339161\pi$
$258$	0	0
$259$	6.60571	0.410459
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.0507	0.989729	0.494865	−	0.868970i	$-0.335218\pi$
$263$	16.0507	0.989729	0.494865	+	0.868970i	$0.335218\pi$
$264$	0	0
$265$	5.91031	0.363067
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.59224	−0.158052	−0.0790258	−	0.996873i	$-0.525181\pi$
$269$	−2.59224	−0.158052	−0.0790258	+	0.996873i	$0.525181\pi$
$270$	0	0
$271$	17.0302	1.03451	0.517257	−	0.855830i	$-0.326953\pi$
$271$	17.0302	1.03451	0.517257	+	0.855830i	$0.326953\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−64.4613	−3.88717
$276$	0	0
$277$	−5.02318	−0.301814	−0.150907	−	0.988548i	$-0.548219\pi$
$277$	−5.02318	−0.301814	−0.150907	+	0.988548i	$0.548219\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.7730	1.35852	0.679260	−	0.733897i	$-0.262301\pi$
$281$	22.7730	1.35852	0.679260	+	0.733897i	$0.262301\pi$
$282$	0	0
$283$	−9.19364	−0.546505	−0.273253	−	0.961942i	$-0.588099\pi$
$283$	−9.19364	−0.546505	−0.273253	+	0.961942i	$0.588099\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.5789	1.56890
$288$	0	0
$289$	−8.64520	−0.508541
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.70629	−0.158103	−0.0790516	−	0.996871i	$-0.525189\pi$
$293$	−2.70629	−0.158103	−0.0790516	+	0.996871i	$0.525189\pi$
$294$	0	0
$295$	15.0576	0.876685
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.599782	0.0346863
$300$	0	0
$301$	15.7199	0.906078
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.28109	−0.130615
$306$	0	0
$307$	23.4524	1.33850	0.669250	−	0.743038i	$-0.266616\pi$
$307$	23.4524	1.33850	0.669250	+	0.743038i	$0.266616\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.4325	−0.761688	−0.380844	−	0.924639i	$-0.624367\pi$
$311$	−13.4325	−0.761688	−0.380844	+	0.924639i	$0.624367\pi$
$312$	0	0
$313$	2.58141	0.145910	0.0729550	−	0.997335i	$-0.476757\pi$
$313$	2.58141	0.145910	0.0729550	+	0.997335i	$0.476757\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−34.3478	−1.92917	−0.964583	−	0.263778i	$-0.915031\pi$
$317$	−34.3478	−1.92917	−0.964583	+	0.263778i	$0.915031\pi$
$318$	0	0
$319$	2.75918	0.154485
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15.3675	0.855068
$324$	0	0
$325$	14.8377	0.823049
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−17.2507	−0.951063
$330$	0	0
$331$	−34.0384	−1.87092	−0.935460	−	0.353431i	$-0.885015\pi$
$331$	−34.0384	−1.87092	−0.935460	+	0.353431i	$0.885015\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−24.9369	−1.36245
$336$	0	0
$337$	−5.78348	−0.315046	−0.157523	−	0.987515i	$-0.550351\pi$
$337$	−5.78348	−0.315046	−0.157523	+	0.987515i	$0.550351\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.4480	−0.619946
$342$	0	0
$343$	−27.0040	−1.45808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.29563	0.391650	0.195825	−	0.980639i	$-0.437262\pi$
$347$	7.29563	0.391650	0.195825	+	0.980639i	$0.437262\pi$
$348$	0	0
$349$	8.45377	0.452520	0.226260	−	0.974067i	$-0.427350\pi$
$349$	8.45377	0.452520	0.226260	+	0.974067i	$0.427350\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−19.2389	−1.02398	−0.511991	−	0.858991i	$-0.671092\pi$
$353$	−19.2389	−1.02398	−0.511991	+	0.858991i	$0.671092\pi$
$354$	0	0
$355$	53.0373	2.81493
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.3588	−1.44394	−0.721972	−	0.691922i	$-0.756764\pi$
$359$	−27.3588	−1.44394	−0.721972	+	0.691922i	$0.756764\pi$
$360$	0	0
$361$	9.26623	0.487696
$362$	0	0
$363$	0	0
$364$	0	0
$365$	26.5621	1.39032
$366$	0	0
$367$	−20.3345	−1.06145	−0.530727	−	0.847543i	$-0.678081\pi$
$367$	−20.3345	−1.06145	−0.530727	+	0.847543i	$0.678081\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.93937	−0.308357
$372$	0	0
$373$	−31.9820	−1.65596	−0.827981	−	0.560756i	$-0.810511\pi$
$373$	−31.9820	−1.65596	−0.827981	+	0.560756i	$0.810511\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.635109	−0.0327098
$378$	0	0
$379$	25.8053	1.32553	0.662765	−	0.748828i	$-0.269383\pi$
$379$	25.8053	1.32553	0.662765	+	0.748828i	$0.269383\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.9866	−0.765781	−0.382891	−	0.923794i	$-0.625071\pi$
$383$	−14.9866	−0.765781	−0.382891	+	0.923794i	$0.625071\pi$
$384$	0	0
$385$	86.6072	4.41391
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−35.5650	−1.80322	−0.901608	−	0.432555i	$-0.857612\pi$
$389$	−35.5650	−1.80322	−0.901608	+	0.432555i	$0.857612\pi$
$390$	0	0
$391$	1.73365	0.0876745
$392$	0	0
$393$	0	0
$394$	0	0
$395$	67.6583	3.40426
$396$	0	0
$397$	19.7793	0.992697	0.496348	−	0.868123i	$-0.334674\pi$
$397$	19.7793	0.992697	0.496348	+	0.868123i	$0.334674\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.88391	0.393704	0.196852	−	0.980433i	$-0.436928\pi$
$401$	7.88391	0.393704	0.196852	+	0.980433i	$0.436928\pi$
$402$	0	0
$403$	2.63511	0.131264
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.41174	0.317818
$408$	0	0
$409$	−36.7794	−1.81862	−0.909312	−	0.416115i	$-0.863391\pi$
$409$	−36.7794	−1.81862	−0.909312	+	0.416115i	$0.863391\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−15.1316	−0.744577
$414$	0	0
$415$	33.6874	1.65365
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.3744	−1.23962	−0.619809	−	0.784753i	$-0.712790\pi$
$419$	−25.3744	−1.23962	−0.619809	+	0.784753i	$0.712790\pi$
$420$	0	0
$421$	−27.7832	−1.35407	−0.677035	−	0.735950i	$-0.736736\pi$
$421$	−27.7832	−1.35407	−0.677035	+	0.735950i	$0.736736\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	42.8880	2.08037
$426$	0	0
$427$	2.29230	0.110932
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.5256	−0.747841	−0.373920	−	0.927461i	$-0.621987\pi$
$431$	−15.5256	−0.747841	−0.373920	+	0.927461i	$0.621987\pi$
$432$	0	0
$433$	16.4033	0.788291	0.394145	−	0.919048i	$-0.371041\pi$
$433$	16.4033	0.788291	0.394145	+	0.919048i	$0.371041\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.18880	0.152541
$438$	0	0
$439$	25.1644	1.20103	0.600516	−	0.799613i	$-0.294962\pi$
$439$	25.1644	1.20103	0.600516	+	0.799613i	$0.294962\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.5446	0.691037	0.345518	−	0.938412i	$-0.387703\pi$
$443$	14.5446	0.691037	0.345518	+	0.938412i	$0.387703\pi$
$444$	0	0
$445$	30.6634	1.45359
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.1553	1.04557	0.522786	−	0.852464i	$-0.324893\pi$
$449$	22.1553	1.04557	0.522786	+	0.852464i	$0.324893\pi$
$450$	0	0
$451$	25.7984	1.21480
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−19.9353	−0.934579
$456$	0	0
$457$	−32.9850	−1.54297	−0.771486	−	0.636246i	$-0.780486\pi$
$457$	−32.9850	−1.54297	−0.771486	+	0.636246i	$0.780486\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	21.5031	1.00150	0.500749	−	0.865593i	$-0.333058\pi$
$461$	21.5031	1.00150	0.500749	+	0.865593i	$0.333058\pi$
$462$	0	0
$463$	9.08268	0.422108	0.211054	−	0.977474i	$-0.432310\pi$
$463$	9.08268	0.422108	0.211054	+	0.977474i	$0.432310\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.9795	−1.20219	−0.601095	−	0.799178i	$-0.705269\pi$
$467$	−25.9795	−1.20219	−0.601095	+	0.799178i	$0.705269\pi$
$468$	0	0
$469$	25.0595	1.15714
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.2583	0.701576
$474$	0	0
$475$	78.8862	3.61955
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.09402	0.232752	0.116376	−	0.993205i	$-0.462872\pi$
$479$	5.09402	0.232752	0.116376	+	0.993205i	$0.462872\pi$
$480$	0	0
$481$	−1.47585	−0.0672932
$482$	0	0
$483$	0	0
$484$	0	0
$485$	47.8847	2.17433
$486$	0	0
$487$	−17.7197	−0.802956	−0.401478	−	0.915869i	$-0.631503\pi$
$487$	−17.7197	−0.802956	−0.401478	+	0.915869i	$0.631503\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.1783	0.865506	0.432753	−	0.901512i	$-0.357542\pi$
$491$	19.1783	0.865506	0.432753	+	0.901512i	$0.357542\pi$
$492$	0	0
$493$	−1.83576	−0.0826786
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−53.2981	−2.39075
$498$	0	0
$499$	12.6703	0.567202	0.283601	−	0.958942i	$-0.408471\pi$
$499$	12.6703	0.567202	0.283601	+	0.958942i	$0.408471\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.51793	0.246032	0.123016	−	0.992405i	$-0.460743\pi$
$503$	5.51793	0.246032	0.123016	+	0.992405i	$0.460743\pi$
$504$	0	0
$505$	−40.4731	−1.80103
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.7735	1.76293	0.881464	−	0.472251i	$-0.156559\pi$
$509$	39.7735	1.76293	0.881464	+	0.472251i	$0.156559\pi$
$510$	0	0
$511$	−26.6927	−1.18081
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.61723	0.423786
$516$	0	0
$517$	−16.7442	−0.736407
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.8222	0.736993	0.368497	−	0.929629i	$-0.379873\pi$
$521$	16.8222	0.736993	0.368497	+	0.929629i	$0.379873\pi$
$522$	0	0
$523$	−19.3488	−0.846062	−0.423031	−	0.906115i	$-0.639034\pi$
$523$	−19.3488	−0.846062	−0.423031	+	0.906115i	$0.639034\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.61670	0.331789
$528$	0	0
$529$	−22.6403	−0.984359
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.93828	−0.257215
$534$	0	0
$535$	20.4608	0.884599
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−56.6220	−2.43888
$540$	0	0
$541$	−36.3519	−1.56289	−0.781446	−	0.623973i	$-0.785518\pi$
$541$	−36.3519	−1.56289	−0.781446	+	0.623973i	$0.785518\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−54.1882	−2.32117
$546$	0	0
$547$	20.0485	0.857210	0.428605	−	0.903492i	$-0.359005\pi$
$547$	20.0485	0.857210	0.428605	+	0.903492i	$0.359005\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.37662	−0.143849
$552$	0	0
$553$	−67.9910	−2.89127
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.5616	0.659364	0.329682	−	0.944092i	$-0.393058\pi$
$557$	15.5616	0.659364	0.329682	+	0.944092i	$0.393058\pi$
$558$	0	0
$559$	−3.51215	−0.148548
$560$	0	0
$561$	0	0
$562$	0	0
$563$	43.0845	1.81580	0.907898	−	0.419192i	$-0.137687\pi$
$563$	43.0845	1.81580	0.907898	+	0.419192i	$0.137687\pi$
$564$	0	0
$565$	25.2602	1.06270
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.6100	−0.989783	−0.494891	−	0.868955i	$-0.664792\pi$
$569$	−23.6100	−0.989783	−0.494891	+	0.868955i	$0.664792\pi$
$570$	0	0
$571$	41.1060	1.72023	0.860117	−	0.510097i	$-0.170391\pi$
$571$	41.1060	1.72023	0.860117	+	0.510097i	$0.170391\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.89940	0.371131
$576$	0	0
$577$	−15.8212	−0.658644	−0.329322	−	0.944218i	$-0.606820\pi$
$577$	−15.8212	−0.658644	−0.329322	+	0.944218i	$0.606820\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−33.8531	−1.40446
$582$	0	0
$583$	−5.76496	−0.238760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	33.7520	1.39310	0.696548	−	0.717510i	$-0.254718\pi$
$587$	33.7520	1.39310	0.696548	+	0.717510i	$0.254718\pi$
$588$	0	0
$589$	14.0098	0.577265
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.4284	−0.879959	−0.439980	−	0.898008i	$-0.645014\pi$
$593$	−21.4284	−0.879959	−0.439980	+	0.898008i	$0.645014\pi$
$594$	0	0
$595$	−57.6222	−2.36228
$596$	0	0
$597$	0	0
$598$	0	0
$599$	23.1311	0.945113	0.472557	−	0.881300i	$-0.343331\pi$
$599$	23.1311	0.945113	0.472557	+	0.881300i	$0.343331\pi$
$600$	0	0
$601$	−17.5192	−0.714624	−0.357312	−	0.933985i	$-0.616307\pi$
$601$	−17.5192	−0.714624	−0.357312	+	0.933985i	$0.616307\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	35.0705	1.42582
$606$	0	0
$607$	−15.8287	−0.642469	−0.321235	−	0.947000i	$-0.604098\pi$
$607$	−15.8287	−0.642469	−0.321235	+	0.947000i	$0.604098\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.85417	0.155923
$612$	0	0
$613$	−33.7136	−1.36168	−0.680841	−	0.732432i	$-0.738385\pi$
$613$	−33.7136	−1.36168	−0.680841	+	0.732432i	$0.738385\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.5334	0.987678	0.493839	−	0.869553i	$-0.335593\pi$
$617$	24.5334	0.987678	0.493839	+	0.869553i	$0.335593\pi$
$618$	0	0
$619$	−34.9934	−1.40650	−0.703252	−	0.710941i	$-0.748269\pi$
$619$	−34.9934	−1.40650	−0.703252	+	0.710941i	$0.748269\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−30.8142	−1.23455
$624$	0	0
$625$	120.969	4.83877
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.26591	−0.170093
$630$	0	0
$631$	−19.0839	−0.759720	−0.379860	−	0.925044i	$-0.624028\pi$
$631$	−19.0839	−0.759720	−0.379860	+	0.925044i	$0.624028\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	24.6818	0.979468
$636$	0	0
$637$	13.0333	0.516397
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.95024	0.156025	0.0780125	−	0.996952i	$-0.475143\pi$
$641$	3.95024	0.156025	0.0780125	+	0.996952i	$0.475143\pi$
$642$	0	0
$643$	25.7889	1.01701	0.508507	−	0.861058i	$-0.330198\pi$
$643$	25.7889	1.01701	0.508507	+	0.861058i	$0.330198\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.6159	0.771182	0.385591	−	0.922670i	$-0.373998\pi$
$647$	19.6159	0.771182	0.385591	+	0.922670i	$0.373998\pi$
$648$	0	0
$649$	−14.6873	−0.576526
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.2016	−1.06448	−0.532241	−	0.846593i	$-0.678650\pi$
$653$	−27.2016	−1.06448	−0.532241	+	0.846593i	$0.678650\pi$
$654$	0	0
$655$	−74.4532	−2.90913
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.7710	−0.964943	−0.482471	−	0.875912i	$-0.660261\pi$
$659$	−24.7710	−0.964943	−0.482471	+	0.875912i	$0.660261\pi$
$660$	0	0
$661$	−41.3384	−1.60788	−0.803939	−	0.594712i	$-0.797266\pi$
$661$	−41.3384	−1.60788	−0.803939	+	0.594712i	$0.797266\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−105.988	−4.11003
$666$	0	0
$667$	−0.380927	−0.0147496
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.22499	0.0858948
$672$	0	0
$673$	12.4022	0.478070	0.239035	−	0.971011i	$-0.423169\pi$
$673$	12.4022	0.478070	0.239035	+	0.971011i	$0.423169\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.88962	0.149490	0.0747452	−	0.997203i	$-0.476186\pi$
$677$	3.88962	0.149490	0.0747452	+	0.997203i	$0.476186\pi$
$678$	0	0
$679$	−48.1202	−1.84668
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−46.6629	−1.78550	−0.892752	−	0.450548i	$-0.851229\pi$
$683$	−46.6629	−1.78550	−0.892752	+	0.450548i	$0.851229\pi$
$684$	0	0
$685$	80.4062	3.07216
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.32698	0.0505539
$690$	0	0
$691$	22.1783	0.843703	0.421851	−	0.906665i	$-0.361380\pi$
$691$	22.1783	0.843703	0.421851	+	0.906665i	$0.361380\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.25395	0.351022
$696$	0	0
$697$	−17.1644	−0.650148
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.2661	1.36975	0.684876	−	0.728660i	$-0.259857\pi$
$701$	36.2661	1.36975	0.684876	+	0.728660i	$0.259857\pi$
$702$	0	0
$703$	−7.84653	−0.295937
$704$	0	0
$705$	0	0
$706$	0	0
$707$	40.6721	1.52963
$708$	0	0
$709$	−24.1429	−0.906704	−0.453352	−	0.891331i	$-0.649772\pi$
$709$	−24.1429	−0.906704	−0.453352	+	0.891331i	$0.649772\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.58049	0.0591899
$714$	0	0
$715$	−19.3499	−0.723644
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.21668	0.0453744	0.0226872	−	0.999743i	$-0.492778\pi$
$719$	1.21668	0.0453744	0.0226872	+	0.999743i	$0.492778\pi$
$720$	0	0
$721$	−9.66451	−0.359925
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.42357	−0.349983
$726$	0	0
$727$	5.19763	0.192769	0.0963847	−	0.995344i	$-0.469272\pi$
$727$	5.19763	0.192769	0.0963847	+	0.995344i	$0.469272\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.1518	−0.375476
$732$	0	0
$733$	2.70658	0.0999700	0.0499850	−	0.998750i	$-0.484083\pi$
$733$	2.70658	0.0999700	0.0499850	+	0.998750i	$0.484083\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	24.3236	0.895972
$738$	0	0
$739$	10.5759	0.389042	0.194521	−	0.980898i	$-0.437685\pi$
$739$	10.5759	0.389042	0.194521	+	0.980898i	$0.437685\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.7651	−1.05529	−0.527645	−	0.849465i	$-0.676925\pi$
$743$	−28.7651	−1.05529	−0.527645	+	0.849465i	$0.676925\pi$
$744$	0	0
$745$	67.5101	2.47338
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20.5614	−0.751298
$750$	0	0
$751$	29.1483	1.06364	0.531818	−	0.846859i	$-0.321509\pi$
$751$	29.1483	1.06364	0.531818	+	0.846859i	$0.321509\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−65.3772	−2.37932
$756$	0	0
$757$	−45.8518	−1.66651	−0.833257	−	0.552886i	$-0.813526\pi$
$757$	−45.8518	−1.66651	−0.833257	+	0.552886i	$0.813526\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.8304	0.827600	0.413800	−	0.910368i	$-0.364201\pi$
$761$	22.8304	0.827600	0.413800	+	0.910368i	$0.364201\pi$
$762$	0	0
$763$	54.4547	1.97139
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.38072	0.122071
$768$	0	0
$769$	−32.3921	−1.16809	−0.584045	−	0.811721i	$-0.698531\pi$
$769$	−32.3921	−1.16809	−0.584045	+	0.811721i	$0.698531\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−30.8359	−1.10909	−0.554546	−	0.832153i	$-0.687108\pi$
$773$	−30.8359	−1.10909	−0.554546	+	0.832153i	$0.687108\pi$
$774$	0	0
$775$	39.0990	1.40448
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31.5715	−1.13116
$780$	0	0
$781$	−51.7330	−1.85115
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−39.4845	−1.40926
$786$	0	0
$787$	−22.4922	−0.801760	−0.400880	−	0.916131i	$-0.631296\pi$
$787$	−22.4922	−0.801760	−0.400880	+	0.916131i	$0.631296\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−25.3844	−0.902564
$792$	0	0
$793$	−0.512149	−0.0181869
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−24.0208	−0.850859	−0.425430	−	0.904991i	$-0.639877\pi$
$797$	−24.0208	−0.850859	−0.425430	+	0.904991i	$0.639877\pi$
$798$	0	0
$799$	11.1404	0.394118
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−25.9089	−0.914304
$804$	0	0
$805$	−11.9568	−0.421422
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.7398	−0.377591	−0.188796	−	0.982016i	$-0.560458\pi$
$809$	−10.7398	−0.377591	−0.188796	+	0.982016i	$0.560458\pi$
$810$	0	0
$811$	4.72701	0.165988	0.0829939	−	0.996550i	$-0.473552\pi$
$811$	4.72701	0.165988	0.0829939	+	0.996550i	$0.473552\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.21701	−0.0426299
$816$	0	0
$817$	−18.6727	−0.653275
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.24899	0.218091	0.109046	−	0.994037i	$-0.465221\pi$
$821$	6.24899	0.218091	0.109046	+	0.994037i	$0.465221\pi$
$822$	0	0
$823$	−22.9062	−0.798459	−0.399230	−	0.916851i	$-0.630722\pi$
$823$	−22.9062	−0.798459	−0.399230	+	0.916851i	$0.630722\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.09930	0.0382264	0.0191132	−	0.999817i	$-0.493916\pi$
$827$	1.09930	0.0382264	0.0191132	+	0.999817i	$0.493916\pi$
$828$	0	0
$829$	20.9781	0.728599	0.364300	−	0.931282i	$-0.381308\pi$
$829$	20.9781	0.728599	0.364300	+	0.931282i	$0.381308\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	37.6722	1.30527
$834$	0	0
$835$	1.19061	0.0412029
$836$	0	0
$837$	0	0
$838$	0	0
$839$	21.1747	0.731031	0.365515	−	0.930805i	$-0.380893\pi$
$839$	21.1747	0.731031	0.365515	+	0.930805i	$0.380893\pi$
$840$	0	0
$841$	−28.5966	−0.986091
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.45396	0.153221
$846$	0	0
$847$	−35.2429	−1.21096
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.885191	−0.0303440
$852$	0	0
$853$	−21.1927	−0.725624	−0.362812	−	0.931862i	$-0.618183\pi$
$853$	−21.1927	−0.725624	−0.362812	+	0.931862i	$0.618183\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.640189	−0.0218685	−0.0109342	−	0.999940i	$-0.503481\pi$
$857$	−0.640189	−0.0218685	−0.0109342	+	0.999940i	$0.503481\pi$
$858$	0	0
$859$	20.2730	0.691705	0.345852	−	0.938289i	$-0.387590\pi$
$859$	20.2730	0.691705	0.345852	+	0.938289i	$0.387590\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.9937	0.850797	0.425398	−	0.905006i	$-0.360134\pi$
$863$	24.9937	0.850797	0.425398	+	0.905006i	$0.360134\pi$
$864$	0	0
$865$	2.16632	0.0736570
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−65.9945	−2.23871
$870$	0	0
$871$	−5.59881	−0.189709
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−196.117	−6.62998
$876$	0	0
$877$	22.2594	0.751646	0.375823	−	0.926691i	$-0.377360\pi$
$877$	22.2594	0.751646	0.375823	+	0.926691i	$0.377360\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−12.6524	−0.426272	−0.213136	−	0.977023i	$-0.568368\pi$
$881$	−12.6524	−0.426272	−0.213136	+	0.977023i	$0.568368\pi$
$882$	0	0
$883$	−4.84806	−0.163150	−0.0815751	−	0.996667i	$-0.525995\pi$
$883$	−4.84806	−0.163150	−0.0815751	+	0.996667i	$0.525995\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.1523	0.609493	0.304747	−	0.952433i	$-0.401428\pi$
$887$	18.1523	0.609493	0.304747	+	0.952433i	$0.401428\pi$
$888$	0	0
$889$	−24.8032	−0.831872
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.4911	0.685708
$894$	0	0
$895$	−46.5792	−1.55697
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.67358	−0.0558171
$900$	0	0
$901$	3.83559	0.127782
$902$	0	0
$903$	0	0
$904$	0	0
$905$	79.5566	2.64455
$906$	0	0
$907$	9.10518	0.302333	0.151166	−	0.988508i	$-0.451697\pi$
$907$	9.10518	0.302333	0.151166	+	0.988508i	$0.451697\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−28.4703	−0.943263	−0.471631	−	0.881796i	$-0.656335\pi$
$911$	−28.4703	−0.943263	−0.471631	+	0.881796i	$0.656335\pi$
$912$	0	0
$913$	−32.8590	−1.08747
$914$	0	0
$915$	0	0
$916$	0	0
$917$	74.8192	2.47075
$918$	0	0
$919$	−48.9506	−1.61473	−0.807366	−	0.590050i	$-0.799108\pi$
$919$	−48.9506	−1.61473	−0.807366	+	0.590050i	$0.799108\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.9079	0.391954
$924$	0	0
$925$	−21.8983	−0.720012
$926$	0	0
$927$	0	0
$928$	0	0
$929$	52.0023	1.70614	0.853071	−	0.521795i	$-0.174738\pi$
$929$	52.0023	1.70614	0.853071	+	0.521795i	$0.174738\pi$
$930$	0	0
$931$	69.2927	2.27098
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−55.9302	−1.82911
$936$	0	0
$937$	−40.2240	−1.31406	−0.657031	−	0.753864i	$-0.728188\pi$
$937$	−40.2240	−1.31406	−0.657031	+	0.753864i	$0.728188\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−36.5473	−1.19141	−0.595704	−	0.803204i	$-0.703127\pi$
$941$	−36.5473	−1.19141	−0.595704	+	0.803204i	$0.703127\pi$
$942$	0	0
$943$	−3.56167	−0.115984
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.6419	−0.930736	−0.465368	−	0.885117i	$-0.654078\pi$
$947$	−28.6419	−0.930736	−0.465368	+	0.885117i	$0.654078\pi$
$948$	0	0
$949$	5.96371	0.193590
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.26429	0.170527	0.0852636	−	0.996358i	$-0.472827\pi$
$953$	5.26429	0.170527	0.0852636	+	0.996358i	$0.472827\pi$
$954$	0	0
$955$	44.4748	1.43917
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−80.8016	−2.60922
$960$	0	0
$961$	−24.0562	−0.776006
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.4018	−0.431420
$966$	0	0
$967$	26.4649	0.851053	0.425526	−	0.904946i	$-0.360089\pi$
$967$	26.4649	0.851053	0.425526	+	0.904946i	$0.360089\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.9428	0.383263	0.191631	−	0.981467i	$-0.438622\pi$
$971$	11.9428	0.383263	0.191631	+	0.981467i	$0.438622\pi$
$972$	0	0
$973$	−9.29945	−0.298127
$974$	0	0
$975$	0	0
$976$	0	0
$977$	21.4328	0.685697	0.342848	−	0.939391i	$-0.388608\pi$
$977$	21.4328	0.685697	0.342848	+	0.939391i	$0.388608\pi$
$978$	0	0
$979$	−29.9094	−0.955907
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.5272	0.750401	0.375200	−	0.926944i	$-0.377574\pi$
$983$	23.5272	0.750401	0.375200	+	0.926944i	$0.377574\pi$
$984$	0	0
$985$	65.1486	2.07581
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.10652	−0.0669836
$990$	0	0
$991$	−18.8427	−0.598558	−0.299279	−	0.954166i	$-0.596746\pi$
$991$	−18.8427	−0.598558	−0.299279	+	0.954166i	$0.596746\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−70.3789	−2.23116
$996$	0	0
$997$	59.1799	1.87425	0.937123	−	0.348999i	$-0.113479\pi$
$997$	59.1799	1.87425	0.937123	+	0.348999i	$0.113479\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4212.2.a.j.1.5		5
3.2	odd	2		4212.2.a.i.1.1		5
9.2	odd	6		468.2.i.b.157.5	✓	10
9.4	even	3		1404.2.i.b.937.1		10
9.5	odd	6		468.2.i.b.313.5	yes	10
9.7	even	3		1404.2.i.b.469.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.i.b.157.5	✓	10	9.2	odd	6
468.2.i.b.313.5	yes	10	9.5	odd	6
1404.2.i.b.469.1		10	9.7	even	3
1404.2.i.b.937.1		10	9.4	even	3
4212.2.a.i.1.1		5	3.2	odd	2
4212.2.a.j.1.5		5	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:	\( N \)	\(=\)	\( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4212.a (trivial)

\(f(q)\)	\(=\)	\(q+4.45396 q^{5} -4.47585 q^{7} -4.34442 q^{11} +1.00000 q^{13} +2.89047 q^{17} +5.31660 q^{19} +0.599782 q^{23} +14.8377 q^{25} -0.635109 q^{29} +2.63511 q^{31} -19.9353 q^{35} -1.47585 q^{37} -5.93828 q^{41} -3.51215 q^{43} +3.85417 q^{47} +13.0333 q^{49} +1.32698 q^{53} -19.3499 q^{55} +3.38072 q^{59} -0.512149 q^{61} +4.45396 q^{65} -5.59881 q^{67} +11.9079 q^{71} +5.96371 q^{73} +19.4450 q^{77} +15.1906 q^{79} +7.56349 q^{83} +12.8740 q^{85} +6.88454 q^{89} -4.47585 q^{91} +23.6799 q^{95} +10.7511 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 7 q^{5} - 2 q^{7} - 5 q^{11} + 5 q^{13} + 13 q^{17} + q^{19} + 8 q^{25} + 12 q^{29} - 2 q^{31} - 5 q^{35} + 13 q^{37} + 4 q^{41} - 8 q^{43} + 7 q^{47} + 15 q^{49} + 19 q^{53} - 21 q^{55} + 11 q^{59}+ \cdots + 4 q^{97}+O(q^{100}) \) 5 * q + 7 * q^5 - 2 * q^7 - 5 * q^11 + 5 * q^13 + 13 * q^17 + q^19 + 8 * q^25 + 12 * q^29 - 2 * q^31 - 5 * q^35 + 13 * q^37 + 4 * q^41 - 8 * q^43 + 7 * q^47 + 15 * q^49 + 19 * q^53 - 21 * q^55 + 11 * q^59 + 7 * q^61 + 7 * q^65 - 2 * q^67 + 29 * q^71 + 19 * q^73 + 7 * q^77 - 5 * q^79 + 24 * q^83 + 9 * q^85 + 19 * q^89 - 2 * q^91 + 10 * q^95 + 4 * q^97

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