LMFDB - Embedded newform 1152.3.g.e.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:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.6
Root			$0.581861 - 1.28897i$ of defining polynomial
Character	$\chi$	$=$	1152.127
Dual form			1152.3.g.e.127.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46308 q^{5} +5.48331i q^{7} +O(q^{10})$	$q+2.46308 q^{5} +5.48331i q^{7} +10.5830i q^{11} -8.96663 q^{13} -16.2399 q^{17} -2.96663i q^{19} -1.46141i q^{23} -18.9333 q^{25} -25.0905 q^{29} -10.5167i q^{31} +13.5058i q^{35} +16.9666 q^{37} -29.0150 q^{41} -34.9666i q^{43} -86.1254i q^{47} +18.9333 q^{49} -80.1976 q^{53} +26.0667i q^{55} +66.4208i q^{59} +0.966630 q^{61} -22.0855 q^{65} +113.800i q^{67} -90.5097i q^{71} +51.7998 q^{73} -58.0299 q^{77} -80.4499i q^{79} +79.9267i q^{83} -40.0000 q^{85} -142.694 q^{89} -49.1669i q^{91} -7.30703i q^{95} -45.8665 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 48 q^{13} + 88 q^{25} + 16 q^{37} - 88 q^{49} - 112 q^{61} - 304 q^{73} - 320 q^{85} + 112 q^{97}+O(q^{100}) $ 8 * q + 48 * q^13 + 88 * q^25 + 16 * q^37 - 88 * q^49 - 112 * q^61 - 304 * q^73 - 320 * q^85 + 112 * 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
$5$	2.46308	0.492615	0.246308	−	0.969192i	$-0.420783\pi$
$5$	2.46308	0.492615	0.246308	+	0.969192i	$0.420783\pi$
$6$	0	0
$7$	5.48331i	0.783331i	0.920108	+	0.391665i	$0.128101\pi$
$7$	5.48331i	0.783331i	−0.920108	+	0.391665i	$0.871899\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	10.5830i	0.962091i	0.876696	+	0.481046i	$0.159743\pi$
$11$	10.5830i	0.962091i	−0.876696	+	0.481046i	$0.840257\pi$
$12$	0	0
$13$	−8.96663	−0.689741	−0.344870	−	0.938650i	$-0.612077\pi$
$13$	−8.96663	−0.689741	−0.344870	+	0.938650i	$0.612077\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−16.2399	−0.955286	−0.477643	−	0.878554i	$-0.658509\pi$
$17$	−16.2399	−0.955286	−0.477643	+	0.878554i	$0.658509\pi$
$18$	0	0
$19$	− 2.96663i	− 0.156138i	−0.996948	−	0.0780692i	$-0.975124\pi$
$19$	− 2.96663i	− 0.156138i	0.996948	−	0.0780692i	$-0.0248755\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1.46141i	− 0.0635394i	−0.999495	−	0.0317697i	$-0.989886\pi$
$23$	− 1.46141i	− 0.0635394i	0.999495	−	0.0317697i	$-0.0101143\pi$
$24$	0	0
$25$	−18.9333	−0.757330
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−25.0905	−0.865189	−0.432595	−	0.901589i	$-0.642402\pi$
$29$	−25.0905	−0.865189	−0.432595	+	0.901589i	$0.642402\pi$
$30$	0	0
$31$	− 10.5167i	− 0.339248i	−0.985509	−	0.169624i	$-0.945745\pi$
$31$	− 10.5167i	− 0.339248i	0.985509	−	0.169624i	$-0.0542553\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13.5058i	0.385881i
$36$	0	0
$37$	16.9666	0.458558	0.229279	−	0.973361i	$-0.426363\pi$
$37$	16.9666	0.458558	0.229279	+	0.973361i	$0.426363\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−29.0150	−0.707682	−0.353841	−	0.935306i	$-0.615125\pi$
$41$	−29.0150	−0.707682	−0.353841	+	0.935306i	$0.615125\pi$
$42$	0	0
$43$	− 34.9666i	− 0.813177i	−0.913611	−	0.406589i	$-0.866718\pi$
$43$	− 34.9666i	− 0.813177i	0.913611	−	0.406589i	$-0.133282\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 86.1254i	− 1.83246i	−0.400657	−	0.916228i	$-0.631218\pi$
$47$	− 86.1254i	− 1.83246i	0.400657	−	0.916228i	$-0.368782\pi$
$48$	0	0
$49$	18.9333	0.386393
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−80.1976	−1.51316	−0.756581	−	0.653900i	$-0.773132\pi$
$53$	−80.1976	−1.51316	−0.756581	+	0.653900i	$0.773132\pi$
$54$	0	0
$55$	26.0667i	0.473941i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	66.4208i	1.12578i	0.826533	+	0.562889i	$0.190310\pi$
$59$	66.4208i	1.12578i	−0.826533	+	0.562889i	$0.809690\pi$
$60$	0	0
$61$	0.966630	0.0158464	0.00792319	−	0.999969i	$-0.497478\pi$
$61$	0.966630	0.0158464	0.00792319	+	0.999969i	$0.497478\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22.0855	−0.339777
$66$	0	0
$67$	113.800i	1.69850i	0.527988	+	0.849252i	$0.322947\pi$
$67$	113.800i	1.69850i	−0.527988	+	0.849252i	$0.677053\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 90.5097i	− 1.27478i	−0.770540	−	0.637392i	$-0.780013\pi$
$71$	− 90.5097i	− 1.27478i	0.770540	−	0.637392i	$-0.219987\pi$
$72$	0	0
$73$	51.7998	0.709586	0.354793	−	0.934945i	$-0.384551\pi$
$73$	51.7998	0.709586	0.354793	+	0.934945i	$0.384551\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−58.0299	−0.753636
$78$	0	0
$79$	− 80.4499i	− 1.01835i	−0.860662	−	0.509177i	$-0.829950\pi$
$79$	− 80.4499i	− 1.01835i	0.860662	−	0.509177i	$-0.170050\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	79.9267i	0.962972i	0.876454	+	0.481486i	$0.159903\pi$
$83$	79.9267i	0.962972i	−0.876454	+	0.481486i	$0.840097\pi$
$84$	0	0
$85$	−40.0000	−0.470588
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−142.694	−1.60330	−0.801652	−	0.597791i	$-0.796045\pi$
$89$	−142.694	−1.60330	−0.801652	+	0.597791i	$0.796045\pi$
$90$	0	0
$91$	− 49.1669i	− 0.540295i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 7.30703i	− 0.0769161i
$96$	0	0
$97$	−45.8665	−0.472851	−0.236425	−	0.971650i	$-0.575976\pi$
$97$	−45.8665	−0.472851	−0.236425	+	0.971650i	$0.575976\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−83.1204	−0.822975	−0.411487	−	0.911415i	$-0.634991\pi$
$101$	−83.1204	−0.822975	−0.411487	+	0.911415i	$0.634991\pi$
$102$	0	0
$103$	106.517i	1.03414i	0.855942	+	0.517071i	$0.172978\pi$
$103$	106.517i	1.03414i	−0.855942	+	0.517071i	$0.827022\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	114.598i	1.07101i	0.844531	+	0.535507i	$0.179879\pi$
$107$	114.598i	1.07101i	−0.844531	+	0.535507i	$0.820121\pi$
$108$	0	0
$109$	94.7664	0.869417	0.434708	−	0.900571i	$-0.356851\pi$
$109$	94.7664	0.869417	0.434708	+	0.900571i	$0.356851\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−201.808	−1.78591	−0.892955	−	0.450146i	$-0.851372\pi$
$113$	−201.808	−1.78591	−0.892955	+	0.450146i	$0.851372\pi$
$114$	0	0
$115$	− 3.59955i	− 0.0313005i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 89.0483i	− 0.748305i
$120$	0	0
$121$	9.00000	0.0743802
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−108.211	−0.865687
$126$	0	0
$127$	− 171.417i	− 1.34974i	−0.737938	−	0.674868i	$-0.764200\pi$
$127$	− 171.417i	− 1.34974i	0.737938	−	0.674868i	$-0.235800\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	169.328i	1.29258i	0.763092	+	0.646290i	$0.223681\pi$
$131$	169.328i	1.29258i	−0.763092	+	0.646290i	$0.776319\pi$
$132$	0	0
$133$	16.2670	0.122308
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.38088	−0.0173787	−0.00868935	−	0.999962i	$-0.502766\pi$
$137$	−2.38088	−0.0173787	−0.00868935	+	0.999962i	$0.502766\pi$
$138$	0	0
$139$	− 209.800i	− 1.50935i	−0.656098	−	0.754675i	$-0.727794\pi$
$139$	− 209.800i	− 1.50935i	0.656098	−	0.754675i	$-0.272206\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 94.8939i	− 0.663594i
$144$	0	0
$145$	−61.7998	−0.426205
$146$	0	0
$147$	0	0
$148$	0	0
$149$	51.7246	0.347145	0.173572	−	0.984821i	$-0.444469\pi$
$149$	51.7246	0.347145	0.173572	+	0.984821i	$0.444469\pi$
$150$	0	0
$151$	227.150i	1.50430i	0.658991	+	0.752151i	$0.270984\pi$
$151$	227.150i	1.50430i	−0.658991	+	0.752151i	$0.729016\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 25.9034i	− 0.167119i
$156$	0	0
$157$	−110.766	−0.705519	−0.352759	−	0.935714i	$-0.614757\pi$
$157$	−110.766	−0.705519	−0.352759	+	0.935714i	$0.614757\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.01335	0.0497724
$162$	0	0
$163$	276.766i	1.69795i	0.528430	+	0.848977i	$0.322781\pi$
$163$	276.766i	1.69795i	−0.528430	+	0.848977i	$0.677219\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 80.2798i	− 0.480717i	−0.970684	−	0.240359i	$-0.922735\pi$
$167$	− 80.2798i	− 0.480717i	0.970684	−	0.240359i	$-0.0772651\pi$
$168$	0	0
$169$	−88.5996	−0.524258
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−124.533	−0.719844	−0.359922	−	0.932982i	$-0.617197\pi$
$173$	−124.533	−0.719844	−0.359922	+	0.932982i	$0.617197\pi$
$174$	0	0
$175$	− 103.817i	− 0.593240i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	66.4208i	0.371066i	0.982638	+	0.185533i	$0.0594012\pi$
$179$	66.4208i	0.371066i	−0.982638	+	0.185533i	$0.940599\pi$
$180$	0	0
$181$	−160.967	−0.889318	−0.444659	−	0.895700i	$-0.646675\pi$
$181$	−160.967	−0.889318	−0.444659	+	0.895700i	$0.646675\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	41.7901	0.225892
$186$	0	0
$187$	− 171.867i	− 0.919072i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 83.2026i	− 0.435616i	−0.975992	−	0.217808i	$-0.930109\pi$
$191$	− 83.2026i	− 0.435616i	0.975992	−	0.217808i	$-0.0698906\pi$
$192$	0	0
$193$	131.666	0.682209	0.341104	−	0.940025i	$-0.389199\pi$
$193$	131.666	0.682209	0.341104	+	0.940025i	$0.389199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−184.566	−0.936885	−0.468442	−	0.883494i	$-0.655185\pi$
$197$	−184.566	−0.936885	−0.468442	+	0.883494i	$0.655185\pi$
$198$	0	0
$199$	168.717i	0.847824i	0.905703	+	0.423912i	$0.139343\pi$
$199$	168.717i	0.847824i	−0.905703	+	0.423912i	$0.860657\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 137.579i	− 0.677729i
$204$	0	0
$205$	−71.4661	−0.348615
$206$	0	0
$207$	0	0
$208$	0	0
$209$	31.3959	0.150219
$210$	0	0
$211$	197.933i	0.938072i	0.883179	+	0.469036i	$0.155399\pi$
$211$	197.933i	0.938072i	−0.883179	+	0.469036i	$0.844601\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 86.1254i	− 0.400583i
$216$	0	0
$217$	57.6663	0.265743
$218$	0	0
$219$	0	0
$220$	0	0
$221$	145.617	0.658900
$222$	0	0
$223$	− 99.6835i	− 0.447011i	−0.974703	−	0.223506i	$-0.928250\pi$
$223$	− 99.6835i	− 0.447011i	0.974703	−	0.223506i	$-0.0717501\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 243.409i	− 1.07229i	−0.844127	−	0.536143i	$-0.819881\pi$
$227$	− 243.409i	− 1.07229i	0.844127	−	0.536143i	$-0.180119\pi$
$228$	0	0
$229$	454.232	1.98355	0.991774	−	0.128002i	$-0.0408562\pi$
$229$	454.232	1.98355	0.991774	+	0.128002i	$0.0408562\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−77.7346	−0.333625	−0.166812	−	0.985989i	$-0.553347\pi$
$233$	−77.7346	−0.333625	−0.166812	+	0.985989i	$0.553347\pi$
$234$	0	0
$235$	− 212.133i	− 0.902696i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	348.886i	1.45977i	0.683568	+	0.729887i	$0.260427\pi$
$239$	348.886i	1.45977i	−0.683568	+	0.729887i	$0.739573\pi$
$240$	0	0
$241$	75.7998	0.314522	0.157261	−	0.987557i	$-0.449734\pi$
$241$	75.7998	0.314522	0.157261	+	0.987557i	$0.449734\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	46.6340	0.190343
$246$	0	0
$247$	26.6007i	0.107695i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	131.733i	0.524834i	0.964955	+	0.262417i	$0.0845197\pi$
$251$	131.733i	0.524834i	−0.964955	+	0.262417i	$0.915480\pi$
$252$	0	0
$253$	15.4661	0.0611307
$254$	0	0
$255$	0	0
$256$	0	0
$257$	305.093	1.18713	0.593565	−	0.804786i	$-0.297720\pi$
$257$	305.093	1.18713	0.593565	+	0.804786i	$0.297720\pi$
$258$	0	0
$259$	93.0334i	0.359202i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	338.656i	1.28767i	0.765166	+	0.643833i	$0.222657\pi$
$263$	338.656i	1.28767i	−0.765166	+	0.643833i	$0.777343\pi$
$264$	0	0
$265$	−197.533	−0.745407
$266$	0	0
$267$	0	0
$268$	0	0
$269$	271.234	1.00830	0.504152	−	0.863615i	$-0.331805\pi$
$269$	271.234	1.00830	0.504152	+	0.863615i	$0.331805\pi$
$270$	0	0
$271$	− 406.749i	− 1.50092i	−0.660916	−	0.750460i	$-0.729832\pi$
$271$	− 406.749i	− 1.50092i	0.660916	−	0.750460i	$-0.270168\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 200.371i	− 0.728621i
$276$	0	0
$277$	−249.234	−0.899760	−0.449880	−	0.893089i	$-0.648533\pi$
$277$	−249.234	−0.899760	−0.449880	+	0.893089i	$0.648533\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	56.9461	0.202655	0.101328	−	0.994853i	$-0.467691\pi$
$281$	56.9461	0.202655	0.101328	+	0.994853i	$0.467691\pi$
$282$	0	0
$283$	− 136.267i	− 0.481509i	−0.970586	−	0.240754i	$-0.922605\pi$
$283$	− 136.267i	− 0.481509i	0.970586	−	0.240754i	$-0.0773948\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 159.098i	− 0.554349i
$288$	0	0
$289$	−25.2670	−0.0874289
$290$	0	0
$291$	0	0
$292$	0	0
$293$	445.324	1.51988	0.759938	−	0.649996i	$-0.225229\pi$
$293$	445.324	1.51988	0.759938	+	0.649996i	$0.225229\pi$
$294$	0	0
$295$	163.600i	0.554575i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.1039i	0.0438257i
$300$	0	0
$301$	191.733	0.636987
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.38088	0.00780617
$306$	0	0
$307$	329.533i	1.07340i	0.843774	+	0.536698i	$0.180329\pi$
$307$	329.533i	1.07340i	−0.843774	+	0.536698i	$0.819671\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	359.116i	1.15471i	0.816492	+	0.577357i	$0.195916\pi$
$311$	359.116i	1.15471i	−0.816492	+	0.577357i	$0.804084\pi$
$312$	0	0
$313$	147.666	0.471777	0.235889	−	0.971780i	$-0.424200\pi$
$313$	147.666	0.471777	0.235889	+	0.971780i	$0.424200\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	233.663	0.737109	0.368554	−	0.929606i	$-0.379853\pi$
$317$	233.663	0.737109	0.368554	+	0.929606i	$0.379853\pi$
$318$	0	0
$319$	− 265.533i	− 0.832391i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	48.1776i	0.149157i
$324$	0	0
$325$	169.768	0.522362
$326$	0	0
$327$	0	0
$328$	0	0
$329$	472.253	1.43542
$330$	0	0
$331$	163.600i	0.494258i	0.968983	+	0.247129i	$0.0794872\pi$
$331$	163.600i	0.494258i	−0.968983	+	0.247129i	$0.920513\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	280.297i	0.836709i
$336$	0	0
$337$	−35.9333	−0.106627	−0.0533134	−	0.998578i	$-0.516978\pi$
$337$	−35.9333	−0.106627	−0.0533134	+	0.998578i	$0.516978\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	111.298	0.326387
$342$	0	0
$343$	372.499i	1.08600i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 584.282i	− 1.68381i	−0.539626	−	0.841905i	$-0.681435\pi$
$347$	− 584.282i	− 1.68381i	0.539626	−	0.841905i	$-0.318565\pi$
$348$	0	0
$349$	−230.766	−0.661222	−0.330611	−	0.943767i	$-0.607255\pi$
$349$	−230.766	−0.661222	−0.330611	+	0.943767i	$0.607255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	50.0166	0.141690	0.0708450	−	0.997487i	$-0.477430\pi$
$353$	50.0166	0.141690	0.0708450	+	0.997487i	$0.477430\pi$
$354$	0	0
$355$	− 222.932i	− 0.627978i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	253.992i	0.707499i	0.935340	+	0.353749i	$0.115093\pi$
$359$	253.992i	0.707499i	−0.935340	+	0.353749i	$0.884907\pi$
$360$	0	0
$361$	352.199	0.975621
$362$	0	0
$363$	0	0
$364$	0	0
$365$	127.587	0.349553
$366$	0	0
$367$	348.682i	0.950088i	0.879962	+	0.475044i	$0.157568\pi$
$367$	348.682i	0.950088i	−0.879962	+	0.475044i	$0.842432\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 439.749i	− 1.18531i
$372$	0	0
$373$	−646.499	−1.73324	−0.866621	−	0.498967i	$-0.833713\pi$
$373$	−646.499	−1.73324	−0.866621	+	0.498967i	$0.833713\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	224.977	0.596756
$378$	0	0
$379$	− 274.433i	− 0.724097i	−0.932159	−	0.362048i	$-0.882078\pi$
$379$	− 274.433i	− 0.724097i	0.932159	−	0.362048i	$-0.117922\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	272.990i	0.712769i	0.934339	+	0.356384i	$0.115991\pi$
$383$	272.990i	0.712769i	−0.934339	+	0.356384i	$0.884009\pi$
$384$	0	0
$385$	−142.932	−0.371252
$386$	0	0
$387$	0	0
$388$	0	0
$389$	412.679	1.06087	0.530436	−	0.847725i	$-0.322028\pi$
$389$	412.679	1.06087	0.530436	+	0.847725i	$0.322028\pi$
$390$	0	0
$391$	23.7330i	0.0606983i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 198.154i	− 0.501656i
$396$	0	0
$397$	384.700	0.969017	0.484508	−	0.874787i	$-0.338999\pi$
$397$	384.700	0.969017	0.484508	+	0.874787i	$0.338999\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	561.466	1.40016	0.700082	−	0.714063i	$-0.253147\pi$
$401$	561.466	1.40016	0.700082	+	0.714063i	$0.253147\pi$
$402$	0	0
$403$	94.2992i	0.233993i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	179.558i	0.441174i
$408$	0	0
$409$	−253.333	−0.619395	−0.309698	−	0.950835i	$-0.600228\pi$
$409$	−253.333	−0.619395	−0.309698	+	0.950835i	$0.600228\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−364.206	−0.881856
$414$	0	0
$415$	196.865i	0.474374i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 622.985i	− 1.48684i	−0.668827	−	0.743418i	$-0.733203\pi$
$419$	− 622.985i	− 1.48684i	0.668827	−	0.743418i	$-0.266797\pi$
$420$	0	0
$421$	487.300	1.15748	0.578741	−	0.815511i	$-0.303544\pi$
$421$	487.300	1.15748	0.578741	+	0.815511i	$0.303544\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	307.473	0.723467
$426$	0	0
$427$	5.30033i	0.0124130i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 176.635i	− 0.409826i	−0.978780	−	0.204913i	$-0.934309\pi$
$431$	− 176.635i	− 0.409826i	0.978780	−	0.204913i	$-0.0656912\pi$
$432$	0	0
$433$	−257.066	−0.593685	−0.296843	−	0.954926i	$-0.595934\pi$
$433$	−257.066	−0.593685	−0.296843	+	0.954926i	$0.595934\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.33545	−0.00992094
$438$	0	0
$439$	− 638.482i	− 1.45440i	−0.686425	−	0.727201i	$-0.740821\pi$
$439$	− 638.482i	− 1.45440i	0.686425	−	0.727201i	$-0.259179\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	104.722i	0.236392i	0.992990	+	0.118196i	$0.0377112\pi$
$443$	104.722i	0.236392i	−0.992990	+	0.118196i	$0.962289\pi$
$444$	0	0
$445$	−351.466	−0.789811
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−219.132	−0.488043	−0.244022	−	0.969770i	$-0.578467\pi$
$449$	−219.132	−0.488043	−0.244022	+	0.969770i	$0.578467\pi$
$450$	0	0
$451$	− 307.066i	− 0.680855i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 121.102i	− 0.266158i
$456$	0	0
$457$	−46.4004	−0.101533	−0.0507664	−	0.998711i	$-0.516166\pi$
$457$	−46.4004	−0.101533	−0.0507664	+	0.998711i	$0.516166\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	517.377	1.12229	0.561146	−	0.827717i	$-0.310361\pi$
$461$	517.377	1.12229	0.561146	+	0.827717i	$0.310361\pi$
$462$	0	0
$463$	− 111.016i	− 0.239776i	−0.992787	−	0.119888i	$-0.961747\pi$
$463$	− 111.016i	− 0.239776i	0.992787	−	0.119888i	$-0.0382535\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	233.934i	0.500930i	0.968126	+	0.250465i	$0.0805835\pi$
$467$	233.934i	0.500930i	−0.968126	+	0.250465i	$0.919416\pi$
$468$	0	0
$469$	−624.000	−1.33049
$470$	0	0
$471$	0	0
$472$	0	0
$473$	370.052	0.782351
$474$	0	0
$475$	56.1680i	0.118248i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	610.185i	1.27387i	0.770916	+	0.636937i	$0.219799\pi$
$479$	610.185i	1.27387i	−0.770916	+	0.636937i	$0.780201\pi$
$480$	0	0
$481$	−152.133	−0.316286
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−112.973	−0.232933
$486$	0	0
$487$	− 104.183i	− 0.213928i	−0.994263	−	0.106964i	$-0.965887\pi$
$487$	− 104.183i	− 0.213928i	0.994263	−	0.106964i	$-0.0341130\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 423.320i	− 0.862159i	−0.902314	−	0.431080i	$-0.858133\pi$
$491$	− 423.320i	− 0.862159i	0.902314	−	0.431080i	$-0.141867\pi$
$492$	0	0
$493$	407.466	0.826503
$494$	0	0
$495$	0	0
$496$	0	0
$497$	496.293	0.998577
$498$	0	0
$499$	126.932i	0.254373i	0.991879	+	0.127187i	$0.0405947\pi$
$499$	126.932i	0.254373i	−0.991879	+	0.127187i	$0.959405\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 274.452i	− 0.545630i	−0.962067	−	0.272815i	$-0.912045\pi$
$503$	− 274.452i	− 0.545630i	0.962067	−	0.272815i	$-0.0879547\pi$
$504$	0	0
$505$	−204.732	−0.405410
$506$	0	0
$507$	0	0
$508$	0	0
$509$	750.416	1.47429	0.737147	−	0.675732i	$-0.236172\pi$
$509$	750.416	1.47429	0.737147	+	0.675732i	$0.236172\pi$
$510$	0	0
$511$	284.034i	0.555840i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	262.359i	0.509434i
$516$	0	0
$517$	911.466	1.76299
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−822.387	−1.57848	−0.789239	−	0.614086i	$-0.789525\pi$
$521$	−822.387	−1.57848	−0.789239	+	0.614086i	$0.789525\pi$
$522$	0	0
$523$	184.366i	0.352516i	0.984344	+	0.176258i	$0.0563993\pi$
$523$	184.366i	0.352516i	−0.984344	+	0.176258i	$0.943601\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	170.789i	0.324079i
$528$	0	0
$529$	526.864	0.995963
$530$	0	0
$531$	0	0
$532$	0	0
$533$	260.167	0.488117
$534$	0	0
$535$	282.265i	0.527598i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	200.371i	0.371745i
$540$	0	0
$541$	−432.967	−0.800308	−0.400154	−	0.916448i	$-0.631043\pi$
$541$	−432.967	−0.800308	−0.400154	+	0.916448i	$0.631043\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	233.417	0.428288
$546$	0	0
$547$	− 557.567i	− 1.01932i	−0.860376	−	0.509659i	$-0.829771\pi$
$547$	− 557.567i	− 1.01932i	0.860376	−	0.509659i	$-0.170229\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	74.4342i	0.135089i
$552$	0	0
$553$	441.132	0.797708
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−656.984	−1.17950	−0.589752	−	0.807584i	$-0.700774\pi$
$557$	−656.984	−1.17950	−0.589752	+	0.807584i	$0.700774\pi$
$558$	0	0
$559$	313.533i	0.560882i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	204.706i	0.363599i	0.983336	+	0.181799i	$0.0581922\pi$
$563$	204.706i	0.363599i	−0.983336	+	0.181799i	$0.941808\pi$
$564$	0	0
$565$	−497.068	−0.879766
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−251.611	−0.442199	−0.221100	−	0.975251i	$-0.570965\pi$
$569$	−251.611	−0.442199	−0.221100	+	0.975251i	$0.570965\pi$
$570$	0	0
$571$	832.198i	1.45744i	0.684812	+	0.728720i	$0.259884\pi$
$571$	832.198i	1.45744i	−0.684812	+	0.728720i	$0.740116\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	27.6692i	0.0481203i
$576$	0	0
$577$	539.132	0.934372	0.467186	−	0.884159i	$-0.345268\pi$
$577$	539.132	0.934372	0.467186	+	0.884159i	$0.345268\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−438.263	−0.754325
$582$	0	0
$583$	− 848.732i	− 1.45580i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	84.6640i	0.144232i	0.997396	+	0.0721159i	$0.0229751\pi$
$587$	84.6640i	0.144232i	−0.997396	+	0.0721159i	$0.977025\pi$
$588$	0	0
$589$	−31.1991	−0.0529696
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−923.717	−1.55770	−0.778851	−	0.627209i	$-0.784197\pi$
$593$	−923.717	−1.55770	−0.778851	+	0.627209i	$0.784197\pi$
$594$	0	0
$595$	− 219.333i	− 0.368626i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	538.674i	0.899288i	0.893208	+	0.449644i	$0.148449\pi$
$599$	538.674i	0.899288i	−0.893208	+	0.449644i	$0.851551\pi$
$600$	0	0
$601$	−164.334	−0.273434	−0.136717	−	0.990610i	$-0.543655\pi$
$601$	−164.334	−0.273434	−0.136717	+	0.990610i	$0.543655\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.1677	0.0366408
$606$	0	0
$607$	368.984i	0.607881i	0.952691	+	0.303941i	$0.0983024\pi$
$607$	368.984i	0.607881i	−0.952691	+	0.303941i	$0.901698\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	772.255i	1.26392i
$612$	0	0
$613$	968.433	1.57982	0.789912	−	0.613220i	$-0.210126\pi$
$613$	968.433	1.57982	0.789912	+	0.613220i	$0.210126\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−53.2682	−0.0863342	−0.0431671	−	0.999068i	$-0.513745\pi$
$617$	−53.2682	−0.0863342	−0.0431671	+	0.999068i	$0.513745\pi$
$618$	0	0
$619$	1066.80i	1.72342i	0.507399	+	0.861711i	$0.330607\pi$
$619$	1066.80i	1.72342i	−0.507399	+	0.861711i	$0.669393\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 782.436i	− 1.25592i
$624$	0	0
$625$	206.800	0.330880
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−275.536	−0.438054
$630$	0	0
$631$	− 468.049i	− 0.741758i	−0.928681	−	0.370879i	$-0.879056\pi$
$631$	− 468.049i	− 0.741758i	0.928681	−	0.370879i	$-0.120944\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 422.212i	− 0.664901i
$636$	0	0
$637$	−169.768	−0.266511
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−603.043	−0.940784	−0.470392	−	0.882458i	$-0.655887\pi$
$641$	−603.043	−0.940784	−0.470392	+	0.882458i	$0.655887\pi$
$642$	0	0
$643$	− 352.633i	− 0.548418i	−0.961670	−	0.274209i	$-0.911584\pi$
$643$	− 352.633i	− 0.548418i	0.961670	−	0.274209i	$-0.0884160\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 897.790i	− 1.38762i	−0.720158	−	0.693810i	$-0.755931\pi$
$647$	− 897.790i	− 1.38762i	0.720158	−	0.693810i	$-0.244069\pi$
$648$	0	0
$649$	−702.932	−1.08310
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−169.459	−0.259508	−0.129754	−	0.991546i	$-0.541419\pi$
$653$	−169.459	−0.259508	−0.129754	+	0.991546i	$0.541419\pi$
$654$	0	0
$655$	417.068i	0.636745i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 36.4864i	− 0.0553663i	−0.999617	−	0.0276832i	$-0.991187\pi$
$659$	− 36.4864i	− 0.0553663i	0.999617	−	0.0276832i	$-0.00881295\pi$
$660$	0	0
$661$	−1038.77	−1.57151	−0.785754	−	0.618539i	$-0.787725\pi$
$661$	−1038.77	−1.57151	−0.785754	+	0.618539i	$0.787725\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	40.0668	0.0602508
$666$	0	0
$667$	36.6674i	0.0549736i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	10.2298i	0.0152457i
$672$	0	0
$673$	−593.600	−0.882020	−0.441010	−	0.897502i	$-0.645380\pi$
$673$	−593.600	−0.882020	−0.441010	+	0.897502i	$0.645380\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−296.455	−0.437895	−0.218948	−	0.975737i	$-0.570262\pi$
$677$	−296.455	−0.437895	−0.218948	+	0.975737i	$0.570262\pi$
$678$	0	0
$679$	− 251.501i	− 0.370398i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1169.67i	1.71255i	0.516520	+	0.856275i	$0.327227\pi$
$683$	1169.67i	1.71255i	−0.516520	+	0.856275i	$0.672773\pi$
$684$	0	0
$685$	−5.86429	−0.00856101
$686$	0	0
$687$	0	0
$688$	0	0
$689$	719.102	1.04369
$690$	0	0
$691$	− 1242.70i	− 1.79841i	−0.437530	−	0.899204i	$-0.644147\pi$
$691$	− 1242.70i	− 1.79841i	0.437530	−	0.899204i	$-0.355853\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 516.753i	− 0.743529i
$696$	0	0
$697$	471.199	0.676039
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1362.28	−1.94333	−0.971666	−	0.236358i	$-0.924046\pi$
$701$	−1362.28	−1.94333	−0.971666	+	0.236358i	$0.924046\pi$
$702$	0	0
$703$	− 50.3337i	− 0.0715984i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 455.776i	− 0.644661i
$708$	0	0
$709$	1261.97	1.77992	0.889962	−	0.456036i	$-0.150731\pi$
$709$	1261.97	1.77992	0.889962	+	0.456036i	$0.150731\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15.3692	−0.0215556
$714$	0	0
$715$	− 233.731i	− 0.326896i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 951.764i	− 1.32373i	−0.749622	−	0.661867i	$-0.769765\pi$
$719$	− 951.764i	− 1.32373i	0.749622	−	0.661867i	$-0.230235\pi$
$720$	0	0
$721$	−584.065	−0.810076
$722$	0	0
$723$	0	0
$724$	0	0
$725$	475.045	0.655234
$726$	0	0
$727$	− 982.383i	− 1.35128i	−0.737230	−	0.675642i	$-0.763867\pi$
$727$	− 982.383i	− 1.35128i	0.737230	−	0.675642i	$-0.236133\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	567.853i	0.776817i
$732$	0	0
$733$	165.699	0.226055	0.113028	−	0.993592i	$-0.463945\pi$
$733$	165.699	0.226055	0.113028	+	0.993592i	$0.463945\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1204.34	−1.63412
$738$	0	0
$739$	248.999i	0.336940i	0.985707	+	0.168470i	$0.0538827\pi$
$739$	248.999i	0.336940i	−0.985707	+	0.168470i	$0.946117\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 538.674i	− 0.724998i	−0.931984	−	0.362499i	$-0.881924\pi$
$743$	− 538.674i	− 0.724998i	0.931984	−	0.362499i	$-0.118076\pi$
$744$	0	0
$745$	127.402	0.171009
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−628.380	−0.838958
$750$	0	0
$751$	979.882i	1.30477i	0.757888	+	0.652385i	$0.226231\pi$
$751$	979.882i	1.30477i	−0.757888	+	0.652385i	$0.773769\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	559.487i	0.741042i
$756$	0	0
$757$	−1151.90	−1.52166	−0.760831	−	0.648950i	$-0.775209\pi$
$757$	−1151.90	−1.52166	−0.760831	+	0.648950i	$0.775209\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−268.064	−0.352253	−0.176126	−	0.984368i	$-0.556357\pi$
$761$	−268.064	−0.352253	−0.176126	+	0.984368i	$0.556357\pi$
$762$	0	0
$763$	519.634i	0.681041i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 595.571i	− 0.776494i
$768$	0	0
$769$	428.467	0.557174	0.278587	−	0.960411i	$-0.410134\pi$
$769$	428.467	0.557174	0.278587	+	0.960411i	$0.410134\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1118.14	−1.44649	−0.723245	−	0.690592i	$-0.757350\pi$
$773$	−1118.14	−1.44649	−0.723245	+	0.690592i	$0.757350\pi$
$774$	0	0
$775$	199.115i	0.256923i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	86.0767i	0.110496i
$780$	0	0
$781$	957.864	1.22646
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−272.826	−0.347549
$786$	0	0
$787$	− 711.832i	− 0.904488i	−0.891894	−	0.452244i	$-0.850624\pi$
$787$	− 711.832i	− 0.904488i	0.891894	−	0.452244i	$-0.149376\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1106.58i	− 1.39896i
$792$	0	0
$793$	−8.66741	−0.0109299
$794$	0	0
$795$	0	0
$796$	0	0
$797$	209.033	0.262274	0.131137	−	0.991364i	$-0.458137\pi$
$797$	209.033	0.262274	0.131137	+	0.991364i	$0.458137\pi$
$798$	0	0
$799$	1398.67i	1.75052i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	548.197i	0.682687i
$804$	0	0
$805$	19.7375	0.0245186
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−581.170	−0.718381	−0.359190	−	0.933264i	$-0.616947\pi$
$809$	−581.170	−0.718381	−0.359190	+	0.933264i	$0.616947\pi$
$810$	0	0
$811$	− 171.432i	− 0.211383i	−0.994399	−	0.105691i	$-0.966294\pi$
$811$	− 171.432i	− 0.211383i	0.994399	−	0.105691i	$-0.0337056\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	681.697i	0.836437i
$816$	0	0
$817$	−103.733	−0.126968
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1134.49	−1.38184	−0.690921	−	0.722931i	$-0.742795\pi$
$821$	−1134.49	−1.38184	−0.690921	+	0.722931i	$0.742795\pi$
$822$	0	0
$823$	− 379.249i	− 0.460812i	−0.973095	−	0.230406i	$-0.925995\pi$
$823$	− 379.249i	− 0.460812i	0.973095	−	0.230406i	$-0.0740055\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1177.33i	1.42362i	0.702373	+	0.711809i	$0.252124\pi$
$827$	1177.33i	1.42362i	−0.702373	+	0.711809i	$0.747876\pi$
$828$	0	0
$829$	−1278.56	−1.54230	−0.771148	−	0.636656i	$-0.780317\pi$
$829$	−1278.56	−1.54230	−0.771148	+	0.636656i	$0.780317\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−307.473	−0.369116
$834$	0	0
$835$	− 197.735i	− 0.236809i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1015.97i	1.21093i	0.795873	+	0.605464i	$0.207012\pi$
$839$	1015.97i	1.21093i	−0.795873	+	0.605464i	$0.792988\pi$
$840$	0	0
$841$	−211.467	−0.251447
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−218.227	−0.258257
$846$	0	0
$847$	49.3498i	0.0582643i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 24.7951i	− 0.0291365i
$852$	0	0
$853$	−1509.43	−1.76956	−0.884778	−	0.466012i	$-0.845690\pi$
$853$	−1509.43	−1.76956	−0.884778	+	0.466012i	$0.845690\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−844.260	−0.985134	−0.492567	−	0.870275i	$-0.663941\pi$
$857$	−844.260	−0.985134	−0.492567	+	0.870275i	$0.663941\pi$
$858$	0	0
$859$	989.231i	1.15161i	0.817588	+	0.575804i	$0.195311\pi$
$859$	989.231i	1.15161i	−0.817588	+	0.575804i	$0.804689\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1680.23i	− 1.94696i	−0.228774	−	0.973480i	$-0.573472\pi$
$863$	− 1680.23i	− 1.94696i	0.228774	−	0.973480i	$-0.426528\pi$
$864$	0	0
$865$	−306.734	−0.354606
$866$	0	0
$867$	0	0
$868$	0	0
$869$	851.402	0.979749
$870$	0	0
$871$	− 1020.40i	− 1.17153i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 593.355i	− 0.678120i
$876$	0	0
$877$	591.899	0.674913	0.337457	−	0.941341i	$-0.390433\pi$
$877$	591.899	0.674913	0.337457	+	0.941341i	$0.390433\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−797.050	−0.904711	−0.452355	−	0.891838i	$-0.649416\pi$
$881$	−797.050	−0.904711	−0.452355	+	0.891838i	$0.649416\pi$
$882$	0	0
$883$	− 276.964i	− 0.313663i	−0.987625	−	0.156831i	$-0.949872\pi$
$883$	− 276.964i	− 0.313663i	0.987625	−	0.156831i	$-0.0501280\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	910.845i	1.02688i	0.858125	+	0.513441i	$0.171630\pi$
$887$	910.845i	1.02688i	−0.858125	+	0.513441i	$0.828370\pi$
$888$	0	0
$889$	939.931	1.05729
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−255.502	−0.286117
$894$	0	0
$895$	163.600i	0.182793i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	263.869i	0.293514i
$900$	0	0
$901$	1302.40	1.44550
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−396.473	−0.438092
$906$	0	0
$907$	1472.83i	1.62385i	0.583763	+	0.811924i	$0.301580\pi$
$907$	1472.83i	1.62385i	−0.583763	+	0.811924i	$0.698420\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 36.5352i	− 0.0401045i	−0.999799	−	0.0200522i	$-0.993617\pi$
$911$	− 36.5352i	− 0.0401045i	0.999799	−	0.0200522i	$-0.00638325\pi$
$912$	0	0
$913$	−845.864	−0.926467
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−928.479	−1.01252
$918$	0	0
$919$	467.150i	0.508324i	0.967162	+	0.254162i	$0.0817996\pi$
$919$	467.150i	0.508324i	−0.967162	+	0.254162i	$0.918200\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	811.567i	0.879270i
$924$	0	0
$925$	−321.234	−0.347280
$926$	0	0
$927$	0	0
$928$	0	0
$929$	762.190	0.820441	0.410220	−	0.911986i	$-0.365452\pi$
$929$	762.190	0.820441	0.410220	+	0.911986i	$0.365452\pi$
$930$	0	0
$931$	− 56.1680i	− 0.0603308i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 423.320i	− 0.452749i
$936$	0	0
$937$	123.335	0.131627	0.0658137	−	0.997832i	$-0.479036\pi$
$937$	123.335	0.131627	0.0658137	+	0.997832i	$0.479036\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	735.638	0.781762	0.390881	−	0.920441i	$-0.372170\pi$
$941$	735.638	0.781762	0.390881	+	0.920441i	$0.372170\pi$
$942$	0	0
$943$	42.4027i	0.0449657i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	549.610i	0.580370i	0.956971	+	0.290185i	$0.0937168\pi$
$947$	549.610i	0.580370i	−0.956971	+	0.290185i	$0.906283\pi$
$948$	0	0
$949$	−464.469	−0.489430
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1217.33	−1.27737	−0.638684	−	0.769469i	$-0.720521\pi$
$953$	−1217.33	−1.27737	−0.638684	+	0.769469i	$0.720521\pi$
$954$	0	0
$955$	− 204.934i	− 0.214591i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 13.0551i	− 0.0136133i
$960$	0	0
$961$	850.399	0.884911
$962$	0	0
$963$	0	0
$964$	0	0
$965$	324.304	0.336066
$966$	0	0
$967$	− 514.616i	− 0.532178i	−0.963949	−	0.266089i	$-0.914269\pi$
$967$	− 514.616i	− 0.532178i	0.963949	−	0.266089i	$-0.0857314\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1248.49i	1.28578i	0.765959	+	0.642889i	$0.222264\pi$
$971$	1248.49i	1.28578i	−0.765959	+	0.642889i	$0.777736\pi$
$972$	0	0
$973$	1150.40	1.18232
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−126.028	−0.128995	−0.0644973	−	0.997918i	$-0.520544\pi$
$977$	−126.028	−0.128995	−0.0644973	+	0.997918i	$0.520544\pi$
$978$	0	0
$979$	− 1510.13i	− 1.54252i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1018.89i	− 1.03651i	−0.855226	−	0.518256i	$-0.826581\pi$
$983$	− 1018.89i	− 1.03651i	0.855226	−	0.518256i	$-0.173419\pi$
$984$	0	0
$985$	−454.601	−0.461524
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−51.1005	−0.0516688
$990$	0	0
$991$	− 859.981i	− 0.867791i	−0.900963	−	0.433895i	$-0.857139\pi$
$991$	− 859.981i	− 0.867791i	0.900963	−	0.433895i	$-0.142861\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	415.562i	0.417651i
$996$	0	0
$997$	1104.43	1.10776	0.553878	−	0.832598i	$-0.313147\pi$
$997$	1104.43	1.10776	0.553878	+	0.832598i	$0.313147\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.g.e.127.6	yes	8
3.2	odd	2	inner	1152.3.g.e.127.4	yes	8
4.3	odd	2	inner	1152.3.g.e.127.5	yes	8
8.3	odd	2		1152.3.g.d.127.3	✓	8
8.5	even	2		1152.3.g.d.127.4	yes	8
12.11	even	2	inner	1152.3.g.e.127.3	yes	8
16.3	odd	4		2304.3.b.s.127.4		8
16.5	even	4		2304.3.b.s.127.5		8
16.11	odd	4		2304.3.b.r.127.6		8
16.13	even	4		2304.3.b.r.127.3		8
24.5	odd	2		1152.3.g.d.127.6	yes	8
24.11	even	2		1152.3.g.d.127.5	yes	8
48.5	odd	4		2304.3.b.s.127.3		8
48.11	even	4		2304.3.b.r.127.4		8
48.29	odd	4		2304.3.b.r.127.5		8
48.35	even	4		2304.3.b.s.127.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.g.d.127.3	✓	8	8.3	odd	2
1152.3.g.d.127.4	yes	8	8.5	even	2
1152.3.g.d.127.5	yes	8	24.11	even	2
1152.3.g.d.127.6	yes	8	24.5	odd	2
1152.3.g.e.127.3	yes	8	12.11	even	2	inner
1152.3.g.e.127.4	yes	8	3.2	odd	2	inner
1152.3.g.e.127.5	yes	8	4.3	odd	2	inner
1152.3.g.e.127.6	yes	8	1.1	even	1	trivial
2304.3.b.r.127.3		8	16.13	even	4
2304.3.b.r.127.4		8	48.11	even	4
2304.3.b.r.127.5		8	48.29	odd	4
2304.3.b.r.127.6		8	16.11	odd	4
2304.3.b.s.127.3		8	48.5	odd	4
2304.3.b.s.127.4		8	16.3	odd	4
2304.3.b.s.127.5		8	16.5	even	4
2304.3.b.s.127.6		8	48.35	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)

\(f(q)\)	\(=\)	\(q+2.46308 q^{5} +5.48331i q^{7} +O(q^{10})\)	\(q+2.46308 q^{5} +5.48331i q^{7} +10.5830i q^{11} -8.96663 q^{13} -16.2399 q^{17} -2.96663i q^{19} -1.46141i q^{23} -18.9333 q^{25} -25.0905 q^{29} -10.5167i q^{31} +13.5058i q^{35} +16.9666 q^{37} -29.0150 q^{41} -34.9666i q^{43} -86.1254i q^{47} +18.9333 q^{49} -80.1976 q^{53} +26.0667i q^{55} +66.4208i q^{59} +0.966630 q^{61} -22.0855 q^{65} +113.800i q^{67} -90.5097i q^{71} +51.7998 q^{73} -58.0299 q^{77} -80.4499i q^{79} +79.9267i q^{83} -40.0000 q^{85} -142.694 q^{89} -49.1669i q^{91} -7.30703i q^{95} -45.8665 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 48 q^{13} + 88 q^{25} + 16 q^{37} - 88 q^{49} - 112 q^{61} - 304 q^{73} - 320 q^{85} + 112 q^{97}+O(q^{100}) \) 8 * q + 48 * q^13 + 88 * q^25 + 16 * q^37 - 88 * q^49 - 112 * q^61 - 304 * q^73 - 320 * q^85 + 112 * q^97

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