LMFDB - Embedded newform 1152.3.g.c.127.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(127,1152)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1152.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1152.127
Dual form			1152.3.g.c.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36433 q^{5} +1.24213i q^{7} +O(q^{10})$	$q-1.36433 q^{5} +1.24213i q^{7} -5.79796i q^{11} +16.3830 q^{13} +5.01086 q^{17} +26.1835i q^{19} -25.1117i q^{23} -23.1386 q^{25} -32.7743 q^{29} -1.01836i q^{31} -1.69466i q^{35} +14.9948 q^{37} +72.5212 q^{41} +33.4922i q^{43} -66.5640i q^{47} +47.4571 q^{49} +54.6513 q^{53} +7.91030i q^{55} -20.5880i q^{59} +111.026 q^{61} -22.3518 q^{65} +60.9540i q^{67} +80.4576i q^{71} +30.0525 q^{73} +7.20179 q^{77} -80.9441i q^{79} +113.958i q^{83} -6.83644 q^{85} +21.0637 q^{89} +20.3498i q^{91} -35.7228i q^{95} +160.594 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{5}+O(q^{10}) $ 8 * q - 16 * q^5	$ 8 q - 16 q^{5} - 48 q^{13} - 16 q^{17} - 8 q^{25} - 80 q^{29} + 16 q^{37} - 80 q^{41} - 88 q^{49} + 176 q^{53} + 272 q^{61} + 160 q^{65} - 16 q^{73} + 320 q^{77} - 32 q^{85} + 240 q^{89} + 400 q^{97}+O(q^{100}) $ 8 * q - 16 * q^5 - 48 * q^13 - 16 * q^17 - 8 * q^25 - 80 * q^29 + 16 * q^37 - 80 * q^41 - 88 * q^49 + 176 * q^53 + 272 * q^61 + 160 * q^65 - 16 * q^73 + 320 * q^77 - 32 * q^85 + 240 * q^89 + 400 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$1$

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	0
$3$	0	0
$4$	0	0
$5$	−1.36433	−0.272865	−0.136433	−	0.990649i	$-0.543564\pi$
$5$	−1.36433	−0.272865	−0.136433	+	0.990649i	$0.543564\pi$
$6$	0	0
$7$	1.24213i	0.177446i	0.996056	+	0.0887232i	$0.0282787\pi$
$7$	1.24213i	0.177446i	−0.996056	+	0.0887232i	$0.971721\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.79796i	− 0.527087i	−0.964647	−	0.263544i	$-0.915109\pi$
$11$	− 5.79796i	− 0.527087i	0.964647	−	0.263544i	$-0.0848913\pi$
$12$	0	0
$13$	16.3830	1.26023	0.630116	−	0.776501i	$-0.283007\pi$
$13$	16.3830	1.26023	0.630116	+	0.776501i	$0.283007\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.01086	0.294756	0.147378	−	0.989080i	$-0.452917\pi$
$17$	5.01086	0.294756	0.147378	+	0.989080i	$0.452917\pi$
$18$	0	0
$19$	26.1835i	1.37808i	0.724725	+	0.689039i	$0.241967\pi$
$19$	26.1835i	1.37808i	−0.724725	+	0.689039i	$0.758033\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 25.1117i	− 1.09181i	−0.837847	−	0.545906i	$-0.816186\pi$
$23$	− 25.1117i	− 1.09181i	0.837847	−	0.545906i	$-0.183814\pi$
$24$	0	0
$25$	−23.1386	−0.925545
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−32.7743	−1.13015	−0.565074	−	0.825040i	$-0.691152\pi$
$29$	−32.7743	−1.13015	−0.565074	+	0.825040i	$0.691152\pi$
$30$	0	0
$31$	− 1.01836i	− 0.0328504i	−0.999865	−	0.0164252i	$-0.994771\pi$
$31$	− 1.01836i	− 0.0328504i	0.999865	−	0.0164252i	$-0.00522854\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.69466i	− 0.0484189i
$36$	0	0
$37$	14.9948	0.405264	0.202632	−	0.979255i	$-0.435050\pi$
$37$	14.9948	0.405264	0.202632	+	0.979255i	$0.435050\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	72.5212	1.76881	0.884405	−	0.466720i	$-0.154565\pi$
$41$	72.5212	1.76881	0.884405	+	0.466720i	$0.154565\pi$
$42$	0	0
$43$	33.4922i	0.778888i	0.921050	+	0.389444i	$0.127333\pi$
$43$	33.4922i	0.778888i	−0.921050	+	0.389444i	$0.872667\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 66.5640i	− 1.41626i	−0.706085	−	0.708128i	$-0.749540\pi$
$47$	− 66.5640i	− 1.41626i	0.706085	−	0.708128i	$-0.250460\pi$
$48$	0	0
$49$	47.4571	0.968513
$50$	0	0
$51$	0	0
$52$	0	0
$53$	54.6513	1.03116	0.515579	−	0.856842i	$-0.327577\pi$
$53$	54.6513	1.03116	0.515579	+	0.856842i	$0.327577\pi$
$54$	0	0
$55$	7.91030i	0.143824i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 20.5880i	− 0.348949i	−0.984662	−	0.174474i	$-0.944177\pi$
$59$	− 20.5880i	− 0.348949i	0.984662	−	0.174474i	$-0.0558226\pi$
$60$	0	0
$61$	111.026	1.82010	0.910050	−	0.414499i	$-0.136043\pi$
$61$	111.026	1.82010	0.910050	+	0.414499i	$0.136043\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22.3518	−0.343873
$66$	0	0
$67$	60.9540i	0.909762i	0.890552	+	0.454881i	$0.150318\pi$
$67$	60.9540i	0.909762i	−0.890552	+	0.454881i	$0.849682\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	80.4576i	1.13320i	0.823991	+	0.566602i	$0.191742\pi$
$71$	80.4576i	1.13320i	−0.823991	+	0.566602i	$0.808258\pi$
$72$	0	0
$73$	30.0525	0.411679	0.205839	−	0.978586i	$-0.434008\pi$
$73$	30.0525	0.411679	0.205839	+	0.978586i	$0.434008\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.20179	0.0935298
$78$	0	0
$79$	− 80.9441i	− 1.02461i	−0.858804	−	0.512304i	$-0.828792\pi$
$79$	− 80.9441i	− 1.02461i	0.858804	−	0.512304i	$-0.171208\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	113.958i	1.37299i	0.727134	+	0.686496i	$0.240852\pi$
$83$	113.958i	1.37299i	−0.727134	+	0.686496i	$0.759148\pi$
$84$	0	0
$85$	−6.83644	−0.0804288
$86$	0	0
$87$	0	0
$88$	0	0
$89$	21.0637	0.236671	0.118335	−	0.992974i	$-0.462244\pi$
$89$	21.0637	0.236671	0.118335	+	0.992974i	$0.462244\pi$
$90$	0	0
$91$	20.3498i	0.223624i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 35.7228i	− 0.376029i
$96$	0	0
$97$	160.594	1.65561	0.827806	−	0.561014i	$-0.189589\pi$
$97$	160.594	1.65561	0.827806	+	0.561014i	$0.189589\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	76.2681	0.755130	0.377565	−	0.925983i	$-0.376761\pi$
$101$	76.2681	0.755130	0.377565	+	0.925983i	$0.376761\pi$
$102$	0	0
$103$	− 182.763i	− 1.77440i	−0.461383	−	0.887201i	$-0.652647\pi$
$103$	− 182.763i	− 1.77440i	0.461383	−	0.887201i	$-0.347353\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 31.8533i	− 0.297694i	−0.988860	−	0.148847i	$-0.952444\pi$
$107$	− 31.8533i	− 0.297694i	0.988860	−	0.148847i	$-0.0475562\pi$
$108$	0	0
$109$	11.3289	0.103935	0.0519676	−	0.998649i	$-0.483451\pi$
$109$	11.3289	0.103935	0.0519676	+	0.998649i	$0.483451\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−49.9587	−0.442113	−0.221056	−	0.975261i	$-0.570950\pi$
$113$	−49.9587	−0.442113	−0.221056	+	0.975261i	$0.570950\pi$
$114$	0	0
$115$	34.2605i	0.297917i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.22412i	0.0523035i
$120$	0	0
$121$	87.3837	0.722179
$122$	0	0
$123$	0	0
$124$	0	0
$125$	65.6767	0.525414
$126$	0	0
$127$	− 208.236i	− 1.63965i	−0.572614	−	0.819825i	$-0.694071\pi$
$127$	− 208.236i	− 1.63965i	0.572614	−	0.819825i	$-0.305929\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	220.549i	1.68358i	0.539804	+	0.841791i	$0.318498\pi$
$131$	220.549i	1.68358i	−0.539804	+	0.841791i	$0.681502\pi$
$132$	0	0
$133$	−32.5231	−0.244535
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.2664	−0.111433	−0.0557167	−	0.998447i	$-0.517744\pi$
$137$	−15.2664	−0.111433	−0.0557167	+	0.998447i	$0.517744\pi$
$138$	0	0
$139$	− 86.7117i	− 0.623825i	−0.950111	−	0.311912i	$-0.899030\pi$
$139$	− 86.7117i	− 0.623825i	0.950111	−	0.311912i	$-0.100970\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 94.9881i	− 0.664252i
$144$	0	0
$145$	44.7148	0.308378
$146$	0	0
$147$	0	0
$148$	0	0
$149$	146.849	0.985561	0.492780	−	0.870154i	$-0.335981\pi$
$149$	146.849	0.985561	0.492780	+	0.870154i	$0.335981\pi$
$150$	0	0
$151$	195.933i	1.29757i	0.760972	+	0.648785i	$0.224723\pi$
$151$	195.933i	1.29757i	−0.760972	+	0.648785i	$0.775277\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.38938i	0.00896374i
$156$	0	0
$157$	4.65454	0.0296468	0.0148234	−	0.999890i	$-0.495281\pi$
$157$	4.65454	0.0296468	0.0148234	+	0.999890i	$0.495281\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	31.1918	0.193738
$162$	0	0
$163$	59.5489i	0.365331i	0.983175	+	0.182665i	$0.0584725\pi$
$163$	59.5489i	0.365331i	−0.983175	+	0.182665i	$0.941528\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	209.012i	1.25157i	0.779996	+	0.625785i	$0.215221\pi$
$167$	209.012i	1.25157i	−0.779996	+	0.625785i	$0.784779\pi$
$168$	0	0
$169$	99.4032	0.588185
$170$	0	0
$171$	0	0
$172$	0	0
$173$	96.7635	0.559326	0.279663	−	0.960098i	$-0.409777\pi$
$173$	96.7635	0.559326	0.279663	+	0.960098i	$0.409777\pi$
$174$	0	0
$175$	− 28.7411i	− 0.164235i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	49.5039i	0.276558i	0.990393	+	0.138279i	$0.0441571\pi$
$179$	49.5039i	0.276558i	−0.990393	+	0.138279i	$0.955843\pi$
$180$	0	0
$181$	141.417	0.781310	0.390655	−	0.920537i	$-0.372248\pi$
$181$	141.417	0.781310	0.390655	+	0.920537i	$0.372248\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−20.4578	−0.110582
$186$	0	0
$187$	− 29.0528i	− 0.155362i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 116.994i	− 0.612533i	−0.951946	−	0.306267i	$-0.900920\pi$
$191$	− 116.994i	− 0.612533i	0.951946	−	0.306267i	$-0.0990799\pi$
$192$	0	0
$193$	−90.7357	−0.470133	−0.235067	−	0.971979i	$-0.575531\pi$
$193$	−90.7357	−0.470133	−0.235067	+	0.971979i	$0.575531\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−380.039	−1.92913	−0.964566	−	0.263840i	$-0.915011\pi$
$197$	−380.039	−1.92913	−0.964566	+	0.263840i	$0.915011\pi$
$198$	0	0
$199$	− 77.6563i	− 0.390233i	−0.980780	−	0.195116i	$-0.937492\pi$
$199$	− 77.6563i	− 0.390233i	0.980780	−	0.195116i	$-0.0625084\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 40.7098i	− 0.200541i
$204$	0	0
$205$	−98.9425	−0.482646
$206$	0	0
$207$	0	0
$208$	0	0
$209$	151.811	0.726367
$210$	0	0
$211$	191.446i	0.907325i	0.891174	+	0.453662i	$0.149883\pi$
$211$	191.446i	0.907325i	−0.891174	+	0.453662i	$0.850117\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 45.6943i	− 0.212531i
$216$	0	0
$217$	1.26493	0.00582919
$218$	0	0
$219$	0	0
$220$	0	0
$221$	82.0930	0.371462
$222$	0	0
$223$	− 168.451i	− 0.755387i	−0.925931	−	0.377693i	$-0.876717\pi$
$223$	− 168.451i	− 0.755387i	0.925931	−	0.377693i	$-0.123283\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 113.516i	− 0.500071i	−0.968237	−	0.250036i	$-0.919558\pi$
$227$	− 113.516i	− 0.500071i	0.968237	−	0.250036i	$-0.0804424\pi$
$228$	0	0
$229$	117.618	0.513615	0.256808	−	0.966463i	$-0.417329\pi$
$229$	117.618	0.513615	0.256808	+	0.966463i	$0.417329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−277.085	−1.18921	−0.594604	−	0.804019i	$-0.702691\pi$
$233$	−277.085	−1.18921	−0.594604	+	0.804019i	$0.702691\pi$
$234$	0	0
$235$	90.8150i	0.386447i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 343.072i	− 1.43545i	−0.696327	−	0.717724i	$-0.745184\pi$
$239$	− 343.072i	− 1.43545i	0.696327	−	0.717724i	$-0.254816\pi$
$240$	0	0
$241$	−328.140	−1.36157	−0.680787	−	0.732481i	$-0.738362\pi$
$241$	−328.140	−1.36157	−0.680787	+	0.732481i	$0.738362\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−64.7470	−0.264273
$246$	0	0
$247$	428.964i	1.73670i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 452.914i	− 1.80444i	−0.431279	−	0.902219i	$-0.641937\pi$
$251$	− 452.914i	− 1.80444i	0.431279	−	0.902219i	$-0.358063\pi$
$252$	0	0
$253$	−145.596	−0.575480
$254$	0	0
$255$	0	0
$256$	0	0
$257$	346.830	1.34953	0.674767	−	0.738031i	$-0.264244\pi$
$257$	346.830	1.34953	0.674767	+	0.738031i	$0.264244\pi$
$258$	0	0
$259$	18.6254i	0.0719127i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	402.440i	1.53019i	0.643917	+	0.765095i	$0.277308\pi$
$263$	402.440i	1.53019i	−0.643917	+	0.765095i	$0.722692\pi$
$264$	0	0
$265$	−74.5622	−0.281367
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−321.562	−1.19540	−0.597699	−	0.801721i	$-0.703918\pi$
$269$	−321.562	−1.19540	−0.597699	+	0.801721i	$0.703918\pi$
$270$	0	0
$271$	456.902i	1.68599i	0.537924	+	0.842993i	$0.319209\pi$
$271$	456.902i	1.68599i	−0.537924	+	0.842993i	$0.680791\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	134.157i	0.487843i
$276$	0	0
$277$	329.543	1.18969	0.594843	−	0.803842i	$-0.297214\pi$
$277$	329.543	1.18969	0.594843	+	0.803842i	$0.297214\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	175.064	0.623005	0.311503	−	0.950245i	$-0.399168\pi$
$281$	175.064	0.623005	0.311503	+	0.950245i	$0.399168\pi$
$282$	0	0
$283$	− 150.298i	− 0.531087i	−0.964099	−	0.265544i	$-0.914449\pi$
$283$	− 150.298i	− 0.531087i	0.964099	−	0.265544i	$-0.0855515\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	90.0804i	0.313869i
$288$	0	0
$289$	−263.891	−0.913119
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−160.435	−0.547561	−0.273781	−	0.961792i	$-0.588274\pi$
$293$	−160.435	−0.547561	−0.273781	+	0.961792i	$0.588274\pi$
$294$	0	0
$295$	28.0887i	0.0952159i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 411.405i	− 1.37594i
$300$	0	0
$301$	−41.6015	−0.138211
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−151.476	−0.496642
$306$	0	0
$307$	− 168.120i	− 0.547621i	−0.961784	−	0.273811i	$-0.911716\pi$
$307$	− 168.120i	− 0.547621i	0.961784	−	0.273811i	$-0.0882841\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 470.376i	− 1.51246i	−0.654305	−	0.756231i	$-0.727039\pi$
$311$	− 470.376i	− 1.51246i	0.654305	−	0.756231i	$-0.272961\pi$
$312$	0	0
$313$	19.4378	0.0621016	0.0310508	−	0.999518i	$-0.490115\pi$
$313$	19.4378	0.0621016	0.0310508	+	0.999518i	$0.490115\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	242.195	0.764021	0.382011	−	0.924158i	$-0.375232\pi$
$317$	242.195	0.764021	0.382011	+	0.924158i	$0.375232\pi$
$318$	0	0
$319$	190.024i	0.595686i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	131.202i	0.406197i
$324$	0	0
$325$	−379.080	−1.16640
$326$	0	0
$327$	0	0
$328$	0	0
$329$	82.6808	0.251309
$330$	0	0
$331$	− 440.951i	− 1.33218i	−0.745872	−	0.666090i	$-0.767967\pi$
$331$	− 440.951i	− 1.33218i	0.745872	−	0.666090i	$-0.232033\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 83.1612i	− 0.248242i
$336$	0	0
$337$	−250.841	−0.744335	−0.372167	−	0.928166i	$-0.621385\pi$
$337$	−250.841	−0.744335	−0.372167	+	0.928166i	$0.621385\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.90443	−0.0173150
$342$	0	0
$343$	119.812i	0.349306i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.1029i	0.0464060i	0.999731	+	0.0232030i	$0.00738641\pi$
$347$	16.1029i	0.0464060i	−0.999731	+	0.0232030i	$0.992614\pi$
$348$	0	0
$349$	−274.843	−0.787516	−0.393758	−	0.919214i	$-0.628825\pi$
$349$	−274.843	−0.787516	−0.393758	+	0.919214i	$0.628825\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−165.428	−0.468634	−0.234317	−	0.972160i	$-0.575285\pi$
$353$	−165.428	−0.468634	−0.234317	+	0.972160i	$0.575285\pi$
$354$	0	0
$355$	− 109.770i	− 0.309212i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	688.519i	1.91788i	0.283607	+	0.958941i	$0.408469\pi$
$359$	688.519i	1.91788i	−0.283607	+	0.958941i	$0.591531\pi$
$360$	0	0
$361$	−324.574	−0.899096
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−41.0015	−0.112333
$366$	0	0
$367$	102.170i	0.278393i	0.990265	+	0.139196i	$0.0444519\pi$
$367$	102.170i	0.278393i	−0.990265	+	0.139196i	$0.955548\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	67.8838i	0.182975i
$372$	0	0
$373$	−294.317	−0.789052	−0.394526	−	0.918885i	$-0.629091\pi$
$373$	−294.317	−0.789052	−0.394526	+	0.918885i	$0.629091\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−536.942	−1.42425
$378$	0	0
$379$	− 81.1923i	− 0.214228i	−0.994247	−	0.107114i	$-0.965839\pi$
$379$	− 81.1923i	− 0.214228i	0.994247	−	0.107114i	$-0.0341610\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	198.838i	0.519160i	0.965722	+	0.259580i	$0.0835841\pi$
$383$	198.838i	0.519160i	−0.965722	+	0.259580i	$0.916416\pi$
$384$	0	0
$385$	−9.82559	−0.0255210
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−368.767	−0.947987	−0.473993	−	0.880528i	$-0.657188\pi$
$389$	−368.767	−0.947987	−0.473993	+	0.880528i	$0.657188\pi$
$390$	0	0
$391$	− 125.831i	− 0.321819i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	110.434i	0.279580i
$396$	0	0
$397$	−114.315	−0.287947	−0.143973	−	0.989582i	$-0.545988\pi$
$397$	−114.315	−0.287947	−0.143973	+	0.989582i	$0.545988\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−39.9083	−0.0995218	−0.0497609	−	0.998761i	$-0.515846\pi$
$401$	−39.9083	−0.0995218	−0.0497609	+	0.998761i	$0.515846\pi$
$402$	0	0
$403$	− 16.6839i	− 0.0413992i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 86.9391i	− 0.213610i
$408$	0	0
$409$	−269.868	−0.659825	−0.329912	−	0.944012i	$-0.607019\pi$
$409$	−269.868	−0.659825	−0.329912	+	0.944012i	$0.607019\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	25.5728	0.0619197
$414$	0	0
$415$	− 155.476i	− 0.374641i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 20.3559i	− 0.0485821i	−0.999705	−	0.0242910i	$-0.992267\pi$
$419$	− 20.3559i	− 0.0485821i	0.999705	−	0.0242910i	$-0.00773283\pi$
$420$	0	0
$421$	−557.905	−1.32519	−0.662595	−	0.748978i	$-0.730545\pi$
$421$	−557.905	−1.32519	−0.662595	+	0.748978i	$0.730545\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−115.944	−0.272810
$426$	0	0
$427$	137.908i	0.322970i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	376.569i	0.873710i	0.899532	+	0.436855i	$0.143908\pi$
$431$	376.569i	0.873710i	−0.899532	+	0.436855i	$0.856092\pi$
$432$	0	0
$433$	602.876	1.39232	0.696162	−	0.717885i	$-0.254890\pi$
$433$	602.876	1.39232	0.696162	+	0.717885i	$0.254890\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	657.510	1.50460
$438$	0	0
$439$	381.087i	0.868080i	0.900894	+	0.434040i	$0.142912\pi$
$439$	381.087i	0.868080i	−0.900894	+	0.434040i	$0.857088\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	599.838i	1.35404i	0.735966	+	0.677018i	$0.236728\pi$
$443$	599.838i	1.35404i	−0.735966	+	0.677018i	$0.763272\pi$
$444$	0	0
$445$	−28.7377	−0.0645791
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−814.240	−1.81345	−0.906726	−	0.421720i	$-0.861427\pi$
$449$	−814.240	−1.81345	−0.906726	+	0.421720i	$0.861427\pi$
$450$	0	0
$451$	− 420.475i	− 0.932317i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 27.7637i	− 0.0610191i
$456$	0	0
$457$	111.281	0.243502	0.121751	−	0.992561i	$-0.461149\pi$
$457$	111.281	0.243502	0.121751	+	0.992561i	$0.461149\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−507.833	−1.10159	−0.550795	−	0.834641i	$-0.685675\pi$
$461$	−507.833	−1.10159	−0.550795	+	0.834641i	$0.685675\pi$
$462$	0	0
$463$	− 397.302i	− 0.858103i	−0.903280	−	0.429052i	$-0.858848\pi$
$463$	− 397.302i	− 0.858103i	0.903280	−	0.429052i	$-0.141152\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 830.195i	− 1.77772i	−0.458179	−	0.888860i	$-0.651498\pi$
$467$	− 830.195i	− 1.77772i	0.458179	−	0.888860i	$-0.348502\pi$
$468$	0	0
$469$	−75.7126	−0.161434
$470$	0	0
$471$	0	0
$472$	0	0
$473$	194.186	0.410542
$474$	0	0
$475$	− 605.849i	− 1.27547i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	146.251i	0.305325i	0.988278	+	0.152662i	$0.0487847\pi$
$479$	146.251i	0.305325i	−0.988278	+	0.152662i	$0.951215\pi$
$480$	0	0
$481$	245.660	0.510727
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−219.103	−0.451759
$486$	0	0
$487$	− 177.070i	− 0.363593i	−0.983336	−	0.181797i	$-0.941809\pi$
$487$	− 177.070i	− 0.363593i	0.983336	−	0.181797i	$-0.0581913\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	94.9463i	0.193373i	0.995315	+	0.0966866i	$0.0308245\pi$
$491$	94.9463i	0.193373i	−0.995315	+	0.0966866i	$0.969176\pi$
$492$	0	0
$493$	−164.227	−0.333118
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−99.9384	−0.201083
$498$	0	0
$499$	− 744.720i	− 1.49243i	−0.665707	−	0.746213i	$-0.731870\pi$
$499$	− 744.720i	− 1.49243i	0.665707	−	0.746213i	$-0.268130\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	578.757i	1.15061i	0.817939	+	0.575305i	$0.195117\pi$
$503$	578.757i	1.15061i	−0.817939	+	0.575305i	$0.804883\pi$
$504$	0	0
$505$	−104.055	−0.206049
$506$	0	0
$507$	0	0
$508$	0	0
$509$	323.101	0.634777	0.317388	−	0.948296i	$-0.397194\pi$
$509$	323.101	0.634777	0.317388	+	0.948296i	$0.397194\pi$
$510$	0	0
$511$	37.3290i	0.0730509i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	249.349i	0.484172i
$516$	0	0
$517$	−385.935	−0.746490
$518$	0	0
$519$	0	0
$520$	0	0
$521$	582.929	1.11887	0.559433	−	0.828875i	$-0.311019\pi$
$521$	582.929	1.11887	0.559433	+	0.828875i	$0.311019\pi$
$522$	0	0
$523$	− 227.111i	− 0.434247i	−0.976144	−	0.217124i	$-0.930332\pi$
$523$	− 227.111i	− 0.434247i	0.976144	−	0.217124i	$-0.0696675\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.10288i	− 0.00968288i
$528$	0	0
$529$	−101.596	−0.192053
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1188.12	2.22911
$534$	0	0
$535$	43.4582i	0.0812303i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 275.154i	− 0.510491i
$540$	0	0
$541$	551.391	1.01921	0.509603	−	0.860409i	$-0.329792\pi$
$541$	551.391	1.01921	0.509603	+	0.860409i	$0.329792\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.4564	−0.0283603
$546$	0	0
$547$	745.659i	1.36318i	0.731735	+	0.681590i	$0.238711\pi$
$547$	745.659i	1.36318i	−0.731735	+	0.681590i	$0.761289\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 858.144i	− 1.55743i
$552$	0	0
$553$	100.543	0.181813
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−755.207	−1.35585	−0.677924	−	0.735132i	$-0.737120\pi$
$557$	−755.207	−1.35585	−0.677924	+	0.735132i	$0.737120\pi$
$558$	0	0
$559$	548.703i	0.981580i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	699.309i	1.24211i	0.783766	+	0.621056i	$0.213296\pi$
$563$	699.309i	1.24211i	−0.783766	+	0.621056i	$0.786704\pi$
$564$	0	0
$565$	68.1600	0.120637
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.8709	−0.0630419	−0.0315210	−	0.999503i	$-0.510035\pi$
$569$	−35.8709	−0.0630419	−0.0315210	+	0.999503i	$0.510035\pi$
$570$	0	0
$571$	828.429i	1.45084i	0.688307	+	0.725420i	$0.258354\pi$
$571$	828.429i	1.45084i	−0.688307	+	0.725420i	$0.741646\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	581.049i	1.01052i
$576$	0	0
$577$	−471.333	−0.816867	−0.408434	−	0.912788i	$-0.633925\pi$
$577$	−471.333	−0.816867	−0.408434	+	0.912788i	$0.633925\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−141.550	−0.243632
$582$	0	0
$583$	− 316.866i	− 0.543510i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	645.149i	1.09906i	0.835473	+	0.549531i	$0.185193\pi$
$587$	645.149i	1.09906i	−0.835473	+	0.549531i	$0.814807\pi$
$588$	0	0
$589$	26.6643	0.0452704
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−203.619	−0.343370	−0.171685	−	0.985152i	$-0.554921\pi$
$593$	−203.619	−0.343370	−0.171685	+	0.985152i	$0.554921\pi$
$594$	0	0
$595$	− 8.49172i	− 0.0142718i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 603.605i	− 1.00769i	−0.863795	−	0.503844i	$-0.831919\pi$
$599$	− 603.605i	− 1.00769i	0.863795	−	0.503844i	$-0.168081\pi$
$600$	0	0
$601$	−626.271	−1.04205	−0.521024	−	0.853542i	$-0.674450\pi$
$601$	−626.271	−1.04205	−0.521024	+	0.853542i	$0.674450\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−119.220	−0.197057
$606$	0	0
$607$	421.012i	0.693595i	0.937940	+	0.346797i	$0.112731\pi$
$607$	421.012i	0.693595i	−0.937940	+	0.346797i	$0.887269\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1090.52i	− 1.78481i
$612$	0	0
$613$	12.9743	0.0211652	0.0105826	−	0.999944i	$-0.496631\pi$
$613$	12.9743	0.0211652	0.0105826	+	0.999944i	$0.496631\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	423.164	0.685842	0.342921	−	0.939364i	$-0.388584\pi$
$617$	423.164	0.685842	0.342921	+	0.939364i	$0.388584\pi$
$618$	0	0
$619$	625.820i	1.01102i	0.862821	+	0.505509i	$0.168695\pi$
$619$	625.820i	1.01102i	−0.862821	+	0.505509i	$0.831305\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	26.1637i	0.0419963i
$624$	0	0
$625$	488.861	0.782178
$626$	0	0
$627$	0	0
$628$	0	0
$629$	75.1367	0.119454
$630$	0	0
$631$	− 690.848i	− 1.09485i	−0.836856	−	0.547423i	$-0.815609\pi$
$631$	− 690.848i	− 1.09485i	0.836856	−	0.547423i	$-0.184391\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	284.101i	0.447403i
$636$	0	0
$637$	777.491	1.22055
$638$	0	0
$639$	0	0
$640$	0	0
$641$	369.160	0.575913	0.287957	−	0.957643i	$-0.407024\pi$
$641$	369.160	0.575913	0.287957	+	0.957643i	$0.407024\pi$
$642$	0	0
$643$	− 666.030i	− 1.03582i	−0.855436	−	0.517909i	$-0.826711\pi$
$643$	− 666.030i	− 1.03582i	0.855436	−	0.517909i	$-0.173289\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 651.886i	− 1.00755i	−0.863835	−	0.503776i	$-0.831944\pi$
$647$	− 651.886i	− 1.00755i	0.863835	−	0.503776i	$-0.168056\pi$
$648$	0	0
$649$	−119.368	−0.183926
$650$	0	0
$651$	0	0
$652$	0	0
$653$	903.324	1.38334	0.691672	−	0.722212i	$-0.256874\pi$
$653$	903.324	1.38334	0.691672	+	0.722212i	$0.256874\pi$
$654$	0	0
$655$	− 300.901i	− 0.459391i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 643.621i	− 0.976664i	−0.872658	−	0.488332i	$-0.837606\pi$
$659$	− 643.621i	− 0.976664i	0.872658	−	0.488332i	$-0.162394\pi$
$660$	0	0
$661$	860.187	1.30134	0.650671	−	0.759360i	$-0.274488\pi$
$661$	860.187	1.30134	0.650671	+	0.759360i	$0.274488\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	44.3722	0.0667250
$666$	0	0
$667$	823.017i	1.23391i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 643.725i	− 0.959351i
$672$	0	0
$673$	866.535	1.28757	0.643785	−	0.765206i	$-0.277363\pi$
$673$	866.535	1.28757	0.643785	+	0.765206i	$0.277363\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1307.26	−1.93095	−0.965477	−	0.260489i	$-0.916116\pi$
$677$	−1307.26	−1.93095	−0.965477	+	0.260489i	$0.916116\pi$
$678$	0	0
$679$	199.478i	0.293783i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 783.569i	− 1.14725i	−0.819120	−	0.573623i	$-0.805538\pi$
$683$	− 783.569i	− 1.14725i	0.819120	−	0.573623i	$-0.194462\pi$
$684$	0	0
$685$	20.8283	0.0304063
$686$	0	0
$687$	0	0
$688$	0	0
$689$	895.354	1.29950
$690$	0	0
$691$	1014.95i	1.46882i	0.678708	+	0.734408i	$0.262540\pi$
$691$	1014.95i	1.46882i	−0.678708	+	0.734408i	$0.737460\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	118.303i	0.170220i
$696$	0	0
$697$	363.394	0.521368
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−957.527	−1.36595	−0.682973	−	0.730444i	$-0.739313\pi$
$701$	−957.527	−1.36595	−0.682973	+	0.730444i	$0.739313\pi$
$702$	0	0
$703$	392.615i	0.558485i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	94.7346i	0.133995i
$708$	0	0
$709$	65.7503	0.0927366	0.0463683	−	0.998924i	$-0.485235\pi$
$709$	65.7503	0.0927366	0.0463683	+	0.998924i	$0.485235\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−25.5728	−0.0358665
$714$	0	0
$715$	129.595i	0.181251i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	573.085i	0.797058i	0.917156	+	0.398529i	$0.130479\pi$
$719$	573.085i	0.797058i	−0.917156	+	0.398529i	$0.869521\pi$
$720$	0	0
$721$	227.015	0.314861
$722$	0	0
$723$	0	0
$724$	0	0
$725$	758.352	1.04600
$726$	0	0
$727$	− 249.632i	− 0.343373i	−0.985152	−	0.171686i	$-0.945078\pi$
$727$	− 249.632i	− 0.343373i	0.985152	−	0.171686i	$-0.0549216\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	167.825i	0.229582i
$732$	0	0
$733$	662.187	0.903393	0.451696	−	0.892172i	$-0.350819\pi$
$733$	662.187	0.903393	0.451696	+	0.892172i	$0.350819\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	353.409	0.479524
$738$	0	0
$739$	98.7372i	0.133609i	0.997766	+	0.0668046i	$0.0212804\pi$
$739$	98.7372i	0.133609i	−0.997766	+	0.0668046i	$0.978720\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 906.520i	− 1.22008i	−0.792370	−	0.610041i	$-0.791153\pi$
$743$	− 906.520i	− 1.22008i	0.792370	−	0.610041i	$-0.208847\pi$
$744$	0	0
$745$	−200.349	−0.268925
$746$	0	0
$747$	0	0
$748$	0	0
$749$	39.5657	0.0528248
$750$	0	0
$751$	− 286.284i	− 0.381204i	−0.981667	−	0.190602i	$-0.938956\pi$
$751$	− 286.284i	− 0.381204i	0.981667	−	0.190602i	$-0.0610440\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 267.316i	− 0.354062i
$756$	0	0
$757$	1162.42	1.53556	0.767781	−	0.640712i	$-0.221361\pi$
$757$	1162.42	1.53556	0.767781	+	0.640712i	$0.221361\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	994.905	1.30737	0.653683	−	0.756769i	$-0.273223\pi$
$761$	994.905	1.30737	0.653683	+	0.756769i	$0.273223\pi$
$762$	0	0
$763$	14.0720i	0.0184429i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 337.293i	− 0.439756i
$768$	0	0
$769$	−614.473	−0.799055	−0.399527	−	0.916721i	$-0.630826\pi$
$769$	−614.473	−0.799055	−0.399527	+	0.916721i	$0.630826\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−927.633	−1.20004	−0.600021	−	0.799984i	$-0.704841\pi$
$773$	−927.633	−1.20004	−0.600021	+	0.799984i	$0.704841\pi$
$774$	0	0
$775$	23.5635i	0.0304045i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1898.86i	2.43756i
$780$	0	0
$781$	466.490	0.597298
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6.35031	−0.00808957
$786$	0	0
$787$	781.920i	0.993545i	0.867881	+	0.496772i	$0.165482\pi$
$787$	781.920i	0.993545i	−0.867881	+	0.496772i	$0.834518\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 62.0550i	− 0.0784513i
$792$	0	0
$793$	1818.94	2.29375
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1160.22	1.45574	0.727868	−	0.685718i	$-0.240511\pi$
$797$	1160.22	1.45574	0.727868	+	0.685718i	$0.240511\pi$
$798$	0	0
$799$	− 333.543i	− 0.417450i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 174.243i	− 0.216991i
$804$	0	0
$805$	−42.5558	−0.0528644
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1512.26	−1.86930	−0.934651	−	0.355568i	$-0.884288\pi$
$809$	−1512.26	−1.86930	−0.934651	+	0.355568i	$0.884288\pi$
$810$	0	0
$811$	1586.92i	1.95674i	0.206858	+	0.978371i	$0.433676\pi$
$811$	1586.92i	1.95674i	−0.206858	+	0.978371i	$0.566324\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 81.2441i	− 0.0996860i
$816$	0	0
$817$	−876.942	−1.07337
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1118.96	1.36292	0.681459	−	0.731857i	$-0.261346\pi$
$821$	1118.96	1.36292	0.681459	+	0.731857i	$0.261346\pi$
$822$	0	0
$823$	− 1628.26i	− 1.97844i	−0.146438	−	0.989220i	$-0.546781\pi$
$823$	− 1628.26i	− 1.97844i	0.146438	−	0.989220i	$-0.453219\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	421.552i	0.509736i	0.966976	+	0.254868i	$0.0820321\pi$
$827$	421.552i	0.509736i	−0.966976	+	0.254868i	$0.917968\pi$
$828$	0	0
$829$	−475.263	−0.573297	−0.286649	−	0.958036i	$-0.592541\pi$
$829$	−475.263	−0.573297	−0.286649	+	0.958036i	$0.592541\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	237.801	0.285475
$834$	0	0
$835$	− 285.161i	− 0.341510i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 653.590i	− 0.779010i	−0.921024	−	0.389505i	$-0.872646\pi$
$839$	− 653.590i	− 0.779010i	0.921024	−	0.389505i	$-0.127354\pi$
$840$	0	0
$841$	233.154	0.277234
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−135.618	−0.160495
$846$	0	0
$847$	108.541i	0.128148i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 376.544i	− 0.442472i
$852$	0	0
$853$	140.493	0.164705	0.0823523	−	0.996603i	$-0.473757\pi$
$853$	140.493	0.164705	0.0823523	+	0.996603i	$0.473757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	562.796	0.656704	0.328352	−	0.944555i	$-0.393507\pi$
$857$	562.796	0.656704	0.328352	+	0.944555i	$0.393507\pi$
$858$	0	0
$859$	− 228.316i	− 0.265792i	−0.991130	−	0.132896i	$-0.957572\pi$
$859$	− 228.316i	− 0.265792i	0.991130	−	0.132896i	$-0.0424277\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	892.187i	1.03382i	0.856040	+	0.516910i	$0.172918\pi$
$863$	892.187i	1.03382i	−0.856040	+	0.516910i	$0.827082\pi$
$864$	0	0
$865$	−132.017	−0.152621
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−469.310	−0.540058
$870$	0	0
$871$	998.611i	1.14651i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	81.5787i	0.0932328i
$876$	0	0
$877$	−1406.66	−1.60395	−0.801974	−	0.597359i	$-0.796217\pi$
$877$	−1406.66	−1.60395	−0.801974	+	0.597359i	$0.796217\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−943.043	−1.07042	−0.535212	−	0.844718i	$-0.679768\pi$
$881$	−943.043	−1.07042	−0.535212	+	0.844718i	$0.679768\pi$
$882$	0	0
$883$	− 1146.63i	− 1.29856i	−0.760549	−	0.649280i	$-0.775070\pi$
$883$	− 1146.63i	− 1.29856i	0.760549	−	0.649280i	$-0.224930\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 894.171i	− 1.00808i	−0.863679	−	0.504042i	$-0.831846\pi$
$887$	− 894.171i	− 1.00808i	0.863679	−	0.504042i	$-0.168154\pi$
$888$	0	0
$889$	258.655	0.290950
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1742.88	1.95171
$894$	0	0
$895$	− 67.5395i	− 0.0754631i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.3761i	0.0371258i
$900$	0	0
$901$	273.850	0.303940
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−192.939	−0.213192
$906$	0	0
$907$	− 1358.60i	− 1.49790i	−0.662626	−	0.748950i	$-0.730558\pi$
$907$	− 1358.60i	− 1.49790i	0.662626	−	0.748950i	$-0.269442\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	804.510i	0.883106i	0.897235	+	0.441553i	$0.145572\pi$
$911$	804.510i	0.883106i	−0.897235	+	0.441553i	$0.854428\pi$
$912$	0	0
$913$	660.726	0.723686
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−273.950	−0.298745
$918$	0	0
$919$	− 1704.73i	− 1.85498i	−0.373849	−	0.927490i	$-0.621962\pi$
$919$	− 1704.73i	− 1.85498i	0.373849	−	0.927490i	$-0.378038\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1318.14i	1.42810i
$924$	0	0
$925$	−346.958	−0.375090
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1351.05	1.45431	0.727154	−	0.686475i	$-0.240843\pi$
$929$	1351.05	1.45431	0.727154	+	0.686475i	$0.240843\pi$
$930$	0	0
$931$	1242.59i	1.33469i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	39.6374i	0.0423930i
$936$	0	0
$937$	672.646	0.717872	0.358936	−	0.933362i	$-0.383140\pi$
$937$	672.646	0.717872	0.358936	+	0.933362i	$0.383140\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−528.671	−0.561818	−0.280909	−	0.959734i	$-0.590636\pi$
$941$	−528.671	−0.561818	−0.280909	+	0.959734i	$0.590636\pi$
$942$	0	0
$943$	− 1821.13i	− 1.93121i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	661.066i	0.698063i	0.937111	+	0.349032i	$0.113489\pi$
$947$	661.066i	0.698063i	−0.937111	+	0.349032i	$0.886511\pi$
$948$	0	0
$949$	492.351	0.518811
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1545.41	−1.62163	−0.810815	−	0.585303i	$-0.800976\pi$
$953$	−1545.41	−1.62163	−0.810815	+	0.585303i	$0.800976\pi$
$954$	0	0
$955$	159.618i	0.167139i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 18.9627i	− 0.0197735i
$960$	0	0
$961$	959.963	0.998921
$962$	0	0
$963$	0	0
$964$	0	0
$965$	123.793	0.128283
$966$	0	0
$967$	161.279i	0.166782i	0.996517	+	0.0833912i	$0.0265751\pi$
$967$	161.279i	0.166782i	−0.996517	+	0.0833912i	$0.973425\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.20412i	0.00432969i	0.999998	+	0.00216484i	$0.000689091\pi$
$971$	4.20412i	0.00432969i	−0.999998	+	0.00216484i	$0.999311\pi$
$972$	0	0
$973$	107.707	0.110696
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1348.19	1.37993	0.689967	−	0.723841i	$-0.257625\pi$
$977$	1348.19	1.37993	0.689967	+	0.723841i	$0.257625\pi$
$978$	0	0
$979$	− 122.126i	− 0.124746i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	984.262i	1.00128i	0.865655	+	0.500642i	$0.166903\pi$
$983$	984.262i	1.00128i	−0.865655	+	0.500642i	$0.833097\pi$
$984$	0	0
$985$	518.497	0.526393
$986$	0	0
$987$	0	0
$988$	0	0
$989$	841.045	0.850399
$990$	0	0
$991$	1013.87i	1.02308i	0.859259	+	0.511541i	$0.170925\pi$
$991$	1013.87i	1.02308i	−0.859259	+	0.511541i	$0.829075\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	105.948i	0.106481i
$996$	0	0
$997$	311.310	0.312246	0.156123	−	0.987738i	$-0.450100\pi$
$997$	311.310	0.312246	0.156123	+	0.987738i	$0.450100\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.g.c.127.6		8
3.2	odd	2		384.3.g.b.127.6	yes	8
4.3	odd	2	inner	1152.3.g.c.127.5		8
8.3	odd	2		1152.3.g.f.127.3		8
8.5	even	2		1152.3.g.f.127.4		8
12.11	even	2		384.3.g.b.127.2	yes	8
16.3	odd	4		2304.3.b.q.127.5		8
16.5	even	4		2304.3.b.q.127.4		8
16.11	odd	4		2304.3.b.t.127.4		8
16.13	even	4		2304.3.b.t.127.5		8
24.5	odd	2		384.3.g.a.127.3	✓	8
24.11	even	2		384.3.g.a.127.7	yes	8
48.5	odd	4		768.3.b.f.127.7		8
48.11	even	4		768.3.b.e.127.3		8
48.29	odd	4		768.3.b.e.127.2		8
48.35	even	4		768.3.b.f.127.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.g.a.127.3	✓	8	24.5	odd	2
384.3.g.a.127.7	yes	8	24.11	even	2
384.3.g.b.127.2	yes	8	12.11	even	2
384.3.g.b.127.6	yes	8	3.2	odd	2
768.3.b.e.127.2		8	48.29	odd	4
768.3.b.e.127.3		8	48.11	even	4
768.3.b.f.127.6		8	48.35	even	4
768.3.b.f.127.7		8	48.5	odd	4
1152.3.g.c.127.5		8	4.3	odd	2	inner
1152.3.g.c.127.6		8	1.1	even	1	trivial
1152.3.g.f.127.3		8	8.3	odd	2
1152.3.g.f.127.4		8	8.5	even	2
2304.3.b.q.127.4		8	16.5	even	4
2304.3.b.q.127.5		8	16.3	odd	4
2304.3.b.t.127.4		8	16.11	odd	4
2304.3.b.t.127.5		8	16.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.36433 q^{5} +1.24213i q^{7} +O(q^{10})\)	\(q-1.36433 q^{5} +1.24213i q^{7} -5.79796i q^{11} +16.3830 q^{13} +5.01086 q^{17} +26.1835i q^{19} -25.1117i q^{23} -23.1386 q^{25} -32.7743 q^{29} -1.01836i q^{31} -1.69466i q^{35} +14.9948 q^{37} +72.5212 q^{41} +33.4922i q^{43} -66.5640i q^{47} +47.4571 q^{49} +54.6513 q^{53} +7.91030i q^{55} -20.5880i q^{59} +111.026 q^{61} -22.3518 q^{65} +60.9540i q^{67} +80.4576i q^{71} +30.0525 q^{73} +7.20179 q^{77} -80.9441i q^{79} +113.958i q^{83} -6.83644 q^{85} +21.0637 q^{89} +20.3498i q^{91} -35.7228i q^{95} +160.594 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{5}+O(q^{10}) \) 8 * q - 16 * q^5	\( 8 q - 16 q^{5} - 48 q^{13} - 16 q^{17} - 8 q^{25} - 80 q^{29} + 16 q^{37} - 80 q^{41} - 88 q^{49} + 176 q^{53} + 272 q^{61} + 160 q^{65} - 16 q^{73} + 320 q^{77} - 32 q^{85} + 240 q^{89} + 400 q^{97}+O(q^{100}) \) 8 * q - 16 * q^5 - 48 * q^13 - 16 * q^17 - 8 * q^25 - 80 * q^29 + 16 * q^37 - 80 * q^41 - 88 * q^49 + 176 * q^53 + 272 * q^61 + 160 * q^65 - 16 * q^73 + 320 * q^77 - 32 * q^85 + 240 * q^89 + 400 * q^97

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