LMFDB - Embedded newform 6048.2.h.b.2591.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (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:	$48.2935231425$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.6
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.b.2591.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96713i q^{5} +1.00000i q^{7} +O(q^{10})$	$q+2.96713i q^{5} +1.00000i q^{7} -4.76028 q^{11} -1.26795 q^{13} -0.378937i q^{17} +5.00000i q^{19} +1.93185 q^{23} -3.80385 q^{25} +4.24264i q^{29} +10.4641i q^{31} -2.96713 q^{35} +2.46410 q^{37} -10.6945i q^{41} +4.73205i q^{43} +3.20736 q^{47} -1.00000 q^{49} -8.76268i q^{53} -14.1244i q^{55} -3.48477 q^{59} -4.92820 q^{61} -3.76217i q^{65} +7.66025i q^{67} +0.240237 q^{71} -0.196152 q^{73} -4.76028i q^{77} +1.80385i q^{79} -7.82894 q^{83} +1.12436 q^{85} +1.93185i q^{89} -1.26795i q^{91} -14.8356 q^{95} -12.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{13} - 72 q^{25} - 8 q^{37} - 8 q^{49} + 16 q^{61} + 40 q^{73} - 88 q^{85} - 48 q^{97}+O(q^{100}) $ 8 * q - 24 * q^13 - 72 * q^25 - 8 * q^37 - 8 * q^49 + 16 * q^61 + 40 * q^73 - 88 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$5$	2.96713i	1.32694i	0.748203	+	0.663470i	$0.230917\pi$
$5$	2.96713i	1.32694i	−0.748203	+	0.663470i	$0.769083\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.76028	−1.43528	−0.717639	−	0.696415i	$-0.754777\pi$
$11$	−4.76028	−1.43528	−0.717639	+	0.696415i	$0.754777\pi$
$12$	0	0
$13$	−1.26795	−0.351666	−0.175833	−	0.984420i	$-0.556262\pi$
$13$	−1.26795	−0.351666	−0.175833	+	0.984420i	$0.556262\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.378937i	− 0.0919058i	−0.998944	−	0.0459529i	$-0.985368\pi$
$17$	− 0.378937i	− 0.0919058i	0.998944	−	0.0459529i	$-0.0146324\pi$
$18$	0	0
$19$	5.00000i	1.14708i	0.819178	+	0.573539i	$0.194430\pi$
$19$	5.00000i	1.14708i	−0.819178	+	0.573539i	$0.805570\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.93185	0.402819	0.201409	−	0.979507i	$-0.435448\pi$
$23$	1.93185	0.402819	0.201409	+	0.979507i	$0.435448\pi$
$24$	0	0
$25$	−3.80385	−0.760770
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.24264i	0.787839i	0.919145	+	0.393919i	$0.128881\pi$
$29$	4.24264i	0.787839i	−0.919145	+	0.393919i	$0.871119\pi$
$30$	0	0
$31$	10.4641i	1.87941i	0.341989	+	0.939704i	$0.388900\pi$
$31$	10.4641i	1.87941i	−0.341989	+	0.939704i	$0.611100\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.96713	−0.501536
$36$	0	0
$37$	2.46410	0.405096	0.202548	−	0.979272i	$-0.435078\pi$
$37$	2.46410	0.405096	0.202548	+	0.979272i	$0.435078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.6945i	− 1.67021i	−0.550094	−	0.835103i	$-0.685408\pi$
$41$	− 10.6945i	− 1.67021i	0.550094	−	0.835103i	$-0.314592\pi$
$42$	0	0
$43$	4.73205i	0.721631i	0.932637	+	0.360815i	$0.117502\pi$
$43$	4.73205i	0.721631i	−0.932637	+	0.360815i	$0.882498\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.20736	0.467842	0.233921	−	0.972256i	$-0.424844\pi$
$47$	3.20736	0.467842	0.233921	+	0.972256i	$0.424844\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.76268i	− 1.20365i	−0.798629	−	0.601824i	$-0.794441\pi$
$53$	− 8.76268i	− 1.20365i	0.798629	−	0.601824i	$-0.205559\pi$
$54$	0	0
$55$	− 14.1244i	− 1.90453i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.48477	−0.453678	−0.226839	−	0.973932i	$-0.572839\pi$
$59$	−3.48477	−0.453678	−0.226839	+	0.973932i	$0.572839\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.76217i	− 0.466639i
$66$	0	0
$67$	7.66025i	0.935849i	0.883768	+	0.467924i	$0.154998\pi$
$67$	7.66025i	0.935849i	−0.883768	+	0.467924i	$0.845002\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.240237	0.0285108	0.0142554	−	0.999898i	$-0.495462\pi$
$71$	0.240237	0.0285108	0.0142554	+	0.999898i	$0.495462\pi$
$72$	0	0
$73$	−0.196152	−0.0229579	−0.0114790	−	0.999934i	$-0.503654\pi$
$73$	−0.196152	−0.0229579	−0.0114790	+	0.999934i	$0.503654\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.76028i	− 0.542484i
$78$	0	0
$79$	1.80385i	0.202949i	0.994838	+	0.101474i	$0.0323560\pi$
$79$	1.80385i	0.202949i	−0.994838	+	0.101474i	$0.967644\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.82894	−0.859338	−0.429669	−	0.902986i	$-0.641370\pi$
$83$	−7.82894	−0.859338	−0.429669	+	0.902986i	$0.641370\pi$
$84$	0	0
$85$	1.12436	0.121953
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.93185i	0.204776i	0.994745	+	0.102388i	$0.0326483\pi$
$89$	1.93185i	0.204776i	−0.994745	+	0.102388i	$0.967352\pi$
$90$	0	0
$91$	− 1.26795i	− 0.132917i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−14.8356	−1.52210
$96$	0	0
$97$	−12.9282	−1.31266	−0.656330	−	0.754474i	$-0.727892\pi$
$97$	−12.9282	−1.31266	−0.656330	+	0.754474i	$0.727892\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.52004i	0.449761i	0.974386	+	0.224880i	$0.0721992\pi$
$101$	4.52004i	0.449761i	−0.974386	+	0.224880i	$0.927801\pi$
$102$	0	0
$103$	3.53590i	0.348402i	0.984710	+	0.174201i	$0.0557343\pi$
$103$	3.53590i	0.348402i	−0.984710	+	0.174201i	$0.944266\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.86422	−0.856936	−0.428468	−	0.903557i	$-0.640947\pi$
$107$	−8.86422	−0.856936	−0.428468	+	0.903557i	$0.640947\pi$
$108$	0	0
$109$	−17.5885	−1.68467	−0.842334	−	0.538955i	$-0.818819\pi$
$109$	−17.5885	−1.68467	−0.842334	+	0.538955i	$0.818819\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 8.48528i	− 0.798228i	−0.916901	−	0.399114i	$-0.869318\pi$
$113$	− 8.48528i	− 0.798228i	0.916901	−	0.399114i	$-0.130682\pi$
$114$	0	0
$115$	5.73205i	0.534516i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.378937	0.0347371
$120$	0	0
$121$	11.6603	1.06002
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.54914i	0.317444i
$126$	0	0
$127$	− 17.6603i	− 1.56709i	−0.621332	−	0.783547i	$-0.713408\pi$
$127$	− 17.6603i	− 1.56709i	0.621332	−	0.783547i	$-0.286592\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.41421	−0.123560	−0.0617802	−	0.998090i	$-0.519678\pi$
$131$	−1.41421	−0.123560	−0.0617802	+	0.998090i	$0.519678\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.48477i	− 0.297724i	−0.988858	−	0.148862i	$-0.952439\pi$
$137$	− 3.48477i	− 0.297724i	0.988858	−	0.148862i	$-0.0475610\pi$
$138$	0	0
$139$	11.3205i	0.960193i	0.877216	+	0.480096i	$0.159398\pi$
$139$	11.3205i	0.960193i	−0.877216	+	0.480096i	$0.840602\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.03579	0.504738
$144$	0	0
$145$	−12.5885	−1.04541
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 19.7990i	− 1.62200i	−0.585049	−	0.810998i	$-0.698925\pi$
$149$	− 19.7990i	− 1.62200i	0.585049	−	0.810998i	$-0.301075\pi$
$150$	0	0
$151$	− 14.9282i	− 1.21484i	−0.794381	−	0.607420i	$-0.792205\pi$
$151$	− 14.9282i	− 1.21484i	0.794381	−	0.607420i	$-0.207795\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−31.0483	−2.49386
$156$	0	0
$157$	6.73205	0.537276	0.268638	−	0.963241i	$-0.413426\pi$
$157$	6.73205	0.537276	0.268638	+	0.963241i	$0.413426\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.93185i	0.152251i
$162$	0	0
$163$	− 18.0526i	− 1.41399i	−0.707221	−	0.706993i	$-0.750051\pi$
$163$	− 18.0526i	− 1.41399i	0.707221	−	0.706993i	$-0.249949\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.9048	1.77243	0.886214	−	0.463276i	$-0.153326\pi$
$167$	22.9048	1.77243	0.886214	+	0.463276i	$0.153326\pi$
$168$	0	0
$169$	−11.3923	−0.876331
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 10.3156i	− 0.784280i	−0.919905	−	0.392140i	$-0.871735\pi$
$173$	− 10.3156i	− 0.784280i	0.919905	−	0.392140i	$-0.128265\pi$
$174$	0	0
$175$	− 3.80385i	− 0.287544i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.9700	0.894683	0.447342	−	0.894363i	$-0.352371\pi$
$179$	11.9700	0.894683	0.447342	+	0.894363i	$0.352371\pi$
$180$	0	0
$181$	24.7846	1.84223	0.921113	−	0.389296i	$-0.127282\pi$
$181$	24.7846	1.84223	0.921113	+	0.389296i	$0.127282\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.31130i	0.537538i
$186$	0	0
$187$	1.80385i	0.131910i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.03768	−0.364514	−0.182257	−	0.983251i	$-0.558340\pi$
$191$	−5.03768	−0.364514	−0.182257	+	0.983251i	$0.558340\pi$
$192$	0	0
$193$	9.32051	0.670905	0.335452	−	0.942057i	$-0.391111\pi$
$193$	9.32051	0.670905	0.335452	+	0.942057i	$0.391111\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20.0764i	− 1.43038i	−0.698928	−	0.715192i	$-0.746339\pi$
$197$	− 20.0764i	− 1.43038i	0.698928	−	0.715192i	$-0.253661\pi$
$198$	0	0
$199$	14.2679i	1.01143i	0.862701	+	0.505714i	$0.168771\pi$
$199$	14.2679i	1.01143i	−0.862701	+	0.505714i	$0.831229\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.24264	−0.297775
$204$	0	0
$205$	31.7321	2.21626
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 23.8014i	− 1.64638i
$210$	0	0
$211$	− 14.5359i	− 1.00069i	−0.865825	−	0.500346i	$-0.833206\pi$
$211$	− 14.5359i	− 1.00069i	0.865825	−	0.500346i	$-0.166794\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−14.0406	−0.957561
$216$	0	0
$217$	−10.4641	−0.710350
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.480473i	0.0323201i
$222$	0	0
$223$	− 24.1244i	− 1.61549i	−0.589535	−	0.807743i	$-0.700689\pi$
$223$	− 24.1244i	− 1.61549i	0.589535	−	0.807743i	$-0.299311\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.6274	−1.50183	−0.750917	−	0.660396i	$-0.770388\pi$
$227$	−22.6274	−1.50183	−0.750917	+	0.660396i	$0.770388\pi$
$228$	0	0
$229$	7.46410	0.493242	0.246621	−	0.969112i	$-0.420680\pi$
$229$	7.46410	0.493242	0.246621	+	0.969112i	$0.420680\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.5558i	0.691536i	0.938320	+	0.345768i	$0.112382\pi$
$233$	10.5558i	0.691536i	−0.938320	+	0.345768i	$0.887618\pi$
$234$	0	0
$235$	9.51666i	0.620798i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.7680	−1.40806	−0.704028	−	0.710173i	$-0.748617\pi$
$239$	−21.7680	−1.40806	−0.704028	+	0.710173i	$0.748617\pi$
$240$	0	0
$241$	17.8038	1.14685	0.573423	−	0.819259i	$-0.305615\pi$
$241$	17.8038	1.14685	0.573423	+	0.819259i	$0.305615\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.96713i	− 0.189563i
$246$	0	0
$247$	− 6.33975i	− 0.403388i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.04008	−0.570605	−0.285303	−	0.958438i	$-0.592094\pi$
$251$	−9.04008	−0.570605	−0.285303	+	0.958438i	$0.592094\pi$
$252$	0	0
$253$	−9.19615	−0.578157
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 20.8714i	− 1.30192i	−0.759110	−	0.650962i	$-0.774366\pi$
$257$	− 20.8714i	− 1.30192i	0.759110	−	0.650962i	$-0.225634\pi$
$258$	0	0
$259$	2.46410i	0.153112i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.4171	−0.642348	−0.321174	−	0.947020i	$-0.604077\pi$
$263$	−10.4171	−0.642348	−0.321174	+	0.947020i	$0.604077\pi$
$264$	0	0
$265$	26.0000	1.59717
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 15.2146i	− 0.927649i	−0.885927	−	0.463825i	$-0.846477\pi$
$269$	− 15.2146i	− 0.927649i	0.885927	−	0.463825i	$-0.153523\pi$
$270$	0	0
$271$	11.8564i	0.720225i	0.932909	+	0.360113i	$0.117262\pi$
$271$	11.8564i	0.720225i	−0.932909	+	0.360113i	$0.882738\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.1074	1.09192
$276$	0	0
$277$	−7.92820	−0.476360	−0.238180	−	0.971221i	$-0.576551\pi$
$277$	−7.92820	−0.476360	−0.238180	+	0.971221i	$0.576551\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	29.6985i	1.77166i	0.464007	+	0.885832i	$0.346411\pi$
$281$	29.6985i	1.77166i	−0.464007	+	0.885832i	$0.653589\pi$
$282$	0	0
$283$	31.4641i	1.87035i	0.354190	+	0.935173i	$0.384756\pi$
$283$	31.4641i	1.87035i	−0.354190	+	0.935173i	$0.615244\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.6945	0.631278
$288$	0	0
$289$	16.8564	0.991553
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 7.62587i	− 0.445508i	−0.974875	−	0.222754i	$-0.928495\pi$
$293$	− 7.62587i	− 0.445508i	0.974875	−	0.222754i	$-0.0715047\pi$
$294$	0	0
$295$	− 10.3397i	− 0.602003i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.44949	−0.141658
$300$	0	0
$301$	−4.73205	−0.272751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 14.6226i	− 0.837288i
$306$	0	0
$307$	9.00000i	0.513657i	0.966457	+	0.256829i	$0.0826776\pi$
$307$	9.00000i	0.513657i	−0.966457	+	0.256829i	$0.917322\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.378937	−0.0214876	−0.0107438	−	0.999942i	$-0.503420\pi$
$311$	−0.378937	−0.0214876	−0.0107438	+	0.999942i	$0.503420\pi$
$312$	0	0
$313$	30.0526	1.69867	0.849336	−	0.527853i	$-0.177003\pi$
$313$	30.0526	1.69867	0.849336	+	0.527853i	$0.177003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.8690i	0.947459i	0.880670	+	0.473729i	$0.157093\pi$
$317$	16.8690i	0.947459i	−0.880670	+	0.473729i	$0.842907\pi$
$318$	0	0
$319$	− 20.1962i	− 1.13077i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.89469	0.105423
$324$	0	0
$325$	4.82309	0.267537
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.20736i	0.176828i
$330$	0	0
$331$	24.7846i	1.36229i	0.732151	+	0.681143i	$0.238517\pi$
$331$	24.7846i	1.36229i	−0.732151	+	0.681143i	$0.761483\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−22.7290	−1.24182
$336$	0	0
$337$	−12.4641	−0.678963	−0.339481	−	0.940613i	$-0.610252\pi$
$337$	−12.4641	−0.678963	−0.339481	+	0.940613i	$0.610252\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 49.8120i	− 2.69747i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.1136	1.07975	0.539876	−	0.841744i	$-0.318471\pi$
$347$	20.1136	1.07975	0.539876	+	0.841744i	$0.318471\pi$
$348$	0	0
$349$	−27.4641	−1.47012	−0.735060	−	0.678002i	$-0.762846\pi$
$349$	−27.4641	−1.47012	−0.735060	+	0.678002i	$0.762846\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.4219i	− 0.980501i	−0.871582	−	0.490250i	$-0.836905\pi$
$353$	− 18.4219i	− 0.980501i	0.871582	−	0.490250i	$-0.163095\pi$
$354$	0	0
$355$	0.712813i	0.0378322i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.2827	−0.701035	−0.350518	−	0.936556i	$-0.613994\pi$
$359$	−13.2827	−0.701035	−0.350518	+	0.936556i	$0.613994\pi$
$360$	0	0
$361$	−6.00000	−0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.582009i	− 0.0304638i
$366$	0	0
$367$	− 14.5167i	− 0.757764i	−0.925445	−	0.378882i	$-0.876309\pi$
$367$	− 14.5167i	− 0.757764i	0.925445	−	0.378882i	$-0.123691\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.76268	0.454936
$372$	0	0
$373$	−12.6603	−0.655523	−0.327762	−	0.944760i	$-0.606294\pi$
$373$	−12.6603	−0.655523	−0.327762	+	0.944760i	$0.606294\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.37945i	− 0.277056i
$378$	0	0
$379$	− 17.8038i	− 0.914522i	−0.889332	−	0.457261i	$-0.848830\pi$
$379$	− 17.8038i	− 0.914522i	0.889332	−	0.457261i	$-0.151170\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.82843	−0.144526	−0.0722629	−	0.997386i	$-0.523022\pi$
$383$	−2.82843	−0.144526	−0.0722629	+	0.997386i	$0.523022\pi$
$384$	0	0
$385$	14.1244	0.719844
$386$	0	0
$387$	0	0
$388$	0	0
$389$	29.3195i	1.48656i	0.668980	+	0.743280i	$0.266731\pi$
$389$	29.3195i	1.48656i	−0.668980	+	0.743280i	$0.733269\pi$
$390$	0	0
$391$	− 0.732051i	− 0.0370214i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.35225	−0.269301
$396$	0	0
$397$	−8.92820	−0.448094	−0.224047	−	0.974578i	$-0.571927\pi$
$397$	−8.92820	−0.448094	−0.224047	+	0.974578i	$0.571927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 15.0759i	− 0.752853i	−0.926446	−	0.376427i	$-0.877153\pi$
$401$	− 15.0759i	− 0.752853i	0.926446	−	0.376427i	$-0.122847\pi$
$402$	0	0
$403$	− 13.2679i	− 0.660924i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.7298	−0.581425
$408$	0	0
$409$	34.5885	1.71029	0.855145	−	0.518390i	$-0.173468\pi$
$409$	34.5885	1.71029	0.855145	+	0.518390i	$0.173468\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 3.48477i	− 0.171474i
$414$	0	0
$415$	− 23.2295i	− 1.14029i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.9086	−1.02145	−0.510726	−	0.859744i	$-0.670623\pi$
$419$	−20.9086	−1.02145	−0.510726	+	0.859744i	$0.670623\pi$
$420$	0	0
$421$	14.3205	0.697939	0.348969	−	0.937134i	$-0.386532\pi$
$421$	14.3205	0.697939	0.348969	+	0.937134i	$0.386532\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.44142i	0.0699191i
$426$	0	0
$427$	− 4.92820i	− 0.238492i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−29.7356	−1.43232	−0.716158	−	0.697938i	$-0.754101\pi$
$431$	−29.7356	−1.43232	−0.716158	+	0.697938i	$0.754101\pi$
$432$	0	0
$433$	−25.2679	−1.21430	−0.607150	−	0.794587i	$-0.707687\pi$
$433$	−25.2679	−1.21430	−0.607150	+	0.794587i	$0.707687\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.65926i	0.462065i
$438$	0	0
$439$	− 10.2487i	− 0.489144i	−0.969631	−	0.244572i	$-0.921352\pi$
$439$	− 10.2487i	− 0.489144i	0.969631	−	0.244572i	$-0.0786475\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.93666	−0.472105	−0.236052	−	0.971740i	$-0.575854\pi$
$443$	−9.93666	−0.472105	−0.236052	+	0.971740i	$0.575854\pi$
$444$	0	0
$445$	−5.73205	−0.271725
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.7733i	1.64105i	0.571607	+	0.820527i	$0.306320\pi$
$449$	34.7733i	1.64105i	−0.571607	+	0.820527i	$0.693680\pi$
$450$	0	0
$451$	50.9090i	2.39721i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.76217	0.176373
$456$	0	0
$457$	−26.1769	−1.22450	−0.612252	−	0.790663i	$-0.709736\pi$
$457$	−26.1769	−1.22450	−0.612252	+	0.790663i	$0.709736\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.3166i	0.946240i	0.880998	+	0.473120i	$0.156872\pi$
$461$	20.3166i	0.946240i	−0.880998	+	0.473120i	$0.843128\pi$
$462$	0	0
$463$	− 12.2487i	− 0.569246i	−0.958640	−	0.284623i	$-0.908132\pi$
$463$	− 12.2487i	− 0.569246i	0.958640	−	0.284623i	$-0.0918684\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.1817	0.841349	0.420674	−	0.907212i	$-0.361793\pi$
$467$	18.1817	0.841349	0.420674	+	0.907212i	$0.361793\pi$
$468$	0	0
$469$	−7.66025	−0.353718
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 22.5259i	− 1.03574i
$474$	0	0
$475$	− 19.0192i	− 0.872662i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.9096	−1.41230	−0.706148	−	0.708064i	$-0.749569\pi$
$479$	−30.9096	−1.41230	−0.706148	+	0.708064i	$0.749569\pi$
$480$	0	0
$481$	−3.12436	−0.142458
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 38.3596i	− 1.74182i
$486$	0	0
$487$	− 16.5885i	− 0.751695i	−0.926682	−	0.375847i	$-0.877352\pi$
$487$	− 16.5885i	− 0.751695i	0.926682	−	0.375847i	$-0.122648\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.4214	−0.605700	−0.302850	−	0.953038i	$-0.597938\pi$
$491$	−13.4214	−0.605700	−0.302850	+	0.953038i	$0.597938\pi$
$492$	0	0
$493$	1.60770	0.0724069
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.240237i	0.0107761i
$498$	0	0
$499$	33.6603i	1.50684i	0.657540	+	0.753420i	$0.271597\pi$
$499$	33.6603i	1.50684i	−0.657540	+	0.753420i	$0.728403\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.4201	0.865897	0.432949	−	0.901419i	$-0.357473\pi$
$503$	19.4201	0.865897	0.432949	+	0.901419i	$0.357473\pi$
$504$	0	0
$505$	−13.4115	−0.596806
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.00429i	− 0.133163i	−0.997781	−	0.0665815i	$-0.978791\pi$
$509$	− 3.00429i	− 0.133163i	0.997781	−	0.0665815i	$-0.0212092\pi$
$510$	0	0
$511$	− 0.196152i	− 0.00867727i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.4915	−0.462309
$516$	0	0
$517$	−15.2679	−0.671484
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 29.0793i	− 1.27399i	−0.770869	−	0.636994i	$-0.780178\pi$
$521$	− 29.0793i	− 1.27399i	0.770869	−	0.636994i	$-0.219822\pi$
$522$	0	0
$523$	28.6603i	1.25323i	0.779331	+	0.626613i	$0.215559\pi$
$523$	28.6603i	1.25323i	−0.779331	+	0.626613i	$0.784441\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.96524	0.172729
$528$	0	0
$529$	−19.2679	−0.837737
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.5601i	0.587354i
$534$	0	0
$535$	− 26.3013i	− 1.13710i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.76028	0.205040
$540$	0	0
$541$	33.1051	1.42330	0.711650	−	0.702534i	$-0.247948\pi$
$541$	33.1051	1.42330	0.711650	+	0.702534i	$0.247948\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 52.1872i	− 2.23545i
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21.2132	−0.903713
$552$	0	0
$553$	−1.80385	−0.0767074
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.5312i	1.50551i	0.658303	+	0.752753i	$0.271274\pi$
$557$	35.5312i	1.50551i	−0.658303	+	0.752753i	$0.728726\pi$
$558$	0	0
$559$	− 6.00000i	− 0.253773i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.4944	−0.821590	−0.410795	−	0.911728i	$-0.634749\pi$
$563$	−19.4944	−0.821590	−0.410795	+	0.911728i	$0.634749\pi$
$564$	0	0
$565$	25.1769	1.05920
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.2795i	0.850159i	0.905156	+	0.425080i	$0.139754\pi$
$569$	20.2795i	0.850159i	−0.905156	+	0.425080i	$0.860246\pi$
$570$	0	0
$571$	− 2.24871i	− 0.0941056i	−0.998892	−	0.0470528i	$-0.985017\pi$
$571$	− 2.24871i	− 0.0941056i	0.998892	−	0.0470528i	$-0.0149829\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.34847	−0.306452
$576$	0	0
$577$	−26.3923	−1.09873	−0.549363	−	0.835584i	$-0.685130\pi$
$577$	−26.3923	−1.09873	−0.549363	+	0.835584i	$0.685130\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 7.82894i	− 0.324799i
$582$	0	0
$583$	41.7128i	1.72757i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.9802	1.36124	0.680619	−	0.732638i	$-0.261711\pi$
$587$	32.9802	1.36124	0.680619	+	0.732638i	$0.261711\pi$
$588$	0	0
$589$	−52.3205	−2.15583
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.3872i	0.919331i	0.888092	+	0.459666i	$0.152031\pi$
$593$	22.3872i	0.919331i	−0.888092	+	0.459666i	$0.847969\pi$
$594$	0	0
$595$	1.12436i	0.0460941i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.14351	−0.332735	−0.166367	−	0.986064i	$-0.553204\pi$
$599$	−8.14351	−0.332735	−0.166367	+	0.986064i	$0.553204\pi$
$600$	0	0
$601$	−47.6603	−1.94410	−0.972051	−	0.234769i	$-0.924567\pi$
$601$	−47.6603	−1.94410	−0.972051	+	0.234769i	$0.924567\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	34.5975i	1.40659i
$606$	0	0
$607$	− 23.1769i	− 0.940722i	−0.882474	−	0.470361i	$-0.844124\pi$
$607$	− 23.1769i	− 0.940722i	0.882474	−	0.470361i	$-0.155876\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.06678	−0.164524
$612$	0	0
$613$	1.58846	0.0641572	0.0320786	−	0.999485i	$-0.489787\pi$
$613$	1.58846	0.0641572	0.0320786	+	0.999485i	$0.489787\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.4543i	0.420874i	0.977607	+	0.210437i	$0.0674887\pi$
$617$	10.4543i	0.420874i	−0.977607	+	0.210437i	$0.932511\pi$
$618$	0	0
$619$	11.3397i	0.455783i	0.973687	+	0.227891i	$0.0731831\pi$
$619$	11.3397i	0.455783i	−0.973687	+	0.227891i	$0.926817\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.93185	−0.0773980
$624$	0	0
$625$	−29.5500	−1.18200
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 0.933740i	− 0.0372307i
$630$	0	0
$631$	22.0526i	0.877899i	0.898512	+	0.438949i	$0.144649\pi$
$631$	22.0526i	0.877899i	−0.898512	+	0.438949i	$0.855351\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	52.4002	2.07944
$636$	0	0
$637$	1.26795	0.0502380
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.72741i	0.305214i	0.988287	+	0.152607i	$0.0487669\pi$
$641$	7.72741i	0.305214i	−0.988287	+	0.152607i	$0.951233\pi$
$642$	0	0
$643$	− 4.85641i	− 0.191518i	−0.995405	−	0.0957590i	$-0.969472\pi$
$643$	− 4.85641i	− 0.191518i	0.995405	−	0.0957590i	$-0.0305278\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.7290	−0.893567	−0.446784	−	0.894642i	$-0.647431\pi$
$647$	−22.7290	−0.893567	−0.446784	+	0.894642i	$0.647431\pi$
$648$	0	0
$649$	16.5885	0.651154
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 39.7738i	− 1.55647i	−0.627973	−	0.778235i	$-0.716115\pi$
$653$	− 39.7738i	− 1.55647i	0.627973	−	0.778235i	$-0.283885\pi$
$654$	0	0
$655$	− 4.19615i	− 0.163957i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.0072	0.467735	0.233867	−	0.972269i	$-0.424862\pi$
$659$	12.0072	0.467735	0.233867	+	0.972269i	$0.424862\pi$
$660$	0	0
$661$	−8.73205	−0.339637	−0.169819	−	0.985475i	$-0.554318\pi$
$661$	−8.73205	−0.339637	−0.169819	+	0.985475i	$0.554318\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 14.8356i	− 0.575301i
$666$	0	0
$667$	8.19615i	0.317356i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	23.4596	0.905649
$672$	0	0
$673$	0.928203	0.0357796	0.0178898	−	0.999840i	$-0.494305\pi$
$673$	0.928203	0.0357796	0.0178898	+	0.999840i	$0.494305\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.2846i	0.472136i	0.971737	+	0.236068i	$0.0758588\pi$
$677$	12.2846i	0.472136i	−0.971737	+	0.236068i	$0.924141\pi$
$678$	0	0
$679$	− 12.9282i	− 0.496139i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.83083	0.261374	0.130687	−	0.991424i	$-0.458282\pi$
$683$	6.83083	0.261374	0.130687	+	0.991424i	$0.458282\pi$
$684$	0	0
$685$	10.3397	0.395061
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.1106i	0.423282i
$690$	0	0
$691$	39.3205i	1.49582i	0.663799	+	0.747911i	$0.268943\pi$
$691$	39.3205i	1.49582i	−0.663799	+	0.747911i	$0.731057\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−33.5894	−1.27412
$696$	0	0
$697$	−4.05256	−0.153502
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.1475i	1.02535i	0.858584	+	0.512673i	$0.171345\pi$
$701$	27.1475i	1.02535i	−0.858584	+	0.512673i	$0.828655\pi$
$702$	0	0
$703$	12.3205i	0.464677i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.52004	−0.169994
$708$	0	0
$709$	−0.660254	−0.0247964	−0.0123982	−	0.999923i	$-0.503947\pi$
$709$	−0.660254	−0.0247964	−0.0123982	+	0.999923i	$0.503947\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.2151i	0.757061i
$714$	0	0
$715$	17.9090i	0.669757i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.9845	1.34199	0.670997	−	0.741460i	$-0.265866\pi$
$719$	35.9845	1.34199	0.670997	+	0.741460i	$0.265866\pi$
$720$	0	0
$721$	−3.53590	−0.131684
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 16.1384i	− 0.599364i
$726$	0	0
$727$	18.9282i	0.702008i	0.936374	+	0.351004i	$0.114160\pi$
$727$	18.9282i	0.702008i	−0.936374	+	0.351004i	$0.885840\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.79315	0.0663221
$732$	0	0
$733$	−36.1962	−1.33694	−0.668468	−	0.743741i	$-0.733050\pi$
$733$	−36.1962	−1.33694	−0.668468	+	0.743741i	$0.733050\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 36.4649i	− 1.34320i
$738$	0	0
$739$	− 1.07180i	− 0.0394267i	−0.999806	−	0.0197133i	$-0.993725\pi$
$739$	− 1.07180i	− 0.0394267i	0.999806	−	0.0197133i	$-0.00627536\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18.0430	0.661934	0.330967	−	0.943642i	$-0.392625\pi$
$743$	18.0430	0.661934	0.330967	+	0.943642i	$0.392625\pi$
$744$	0	0
$745$	58.7461	2.15229
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8.86422i	− 0.323892i
$750$	0	0
$751$	− 3.66025i	− 0.133565i	−0.997768	−	0.0667823i	$-0.978727\pi$
$751$	− 3.66025i	− 0.133565i	0.997768	−	0.0667823i	$-0.0212733\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	44.2939	1.61202
$756$	0	0
$757$	52.6410	1.91327	0.956635	−	0.291289i	$-0.0940841\pi$
$757$	52.6410	1.91327	0.956635	+	0.291289i	$0.0940841\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 30.2533i	− 1.09668i	−0.836255	−	0.548340i	$-0.815260\pi$
$761$	− 30.2533i	− 1.09668i	0.836255	−	0.548340i	$-0.184740\pi$
$762$	0	0
$763$	− 17.5885i	− 0.636745i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.41851	0.159543
$768$	0	0
$769$	0.339746	0.0122516	0.00612578	−	0.999981i	$-0.498050\pi$
$769$	0.339746	0.0122516	0.00612578	+	0.999981i	$0.498050\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.5630i	0.811536i	0.913976	+	0.405768i	$0.132996\pi$
$773$	22.5630i	0.811536i	−0.913976	+	0.405768i	$0.867004\pi$
$774$	0	0
$775$	− 39.8038i	− 1.42980i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	53.4727	1.91586
$780$	0	0
$781$	−1.14359	−0.0409210
$782$	0	0
$783$	0	0
$784$	0	0
$785$	19.9749i	0.712933i
$786$	0	0
$787$	− 27.3205i	− 0.973871i	−0.873438	−	0.486935i	$-0.838115\pi$
$787$	− 27.3205i	− 0.973871i	0.873438	−	0.486935i	$-0.161885\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.48528	0.301702
$792$	0	0
$793$	6.24871	0.221898
$794$	0	0
$795$	0	0
$796$	0	0
$797$	45.9483i	1.62757i	0.581164	+	0.813787i	$0.302598\pi$
$797$	45.9483i	1.62757i	−0.581164	+	0.813787i	$0.697402\pi$
$798$	0	0
$799$	− 1.21539i	− 0.0429974i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.933740	0.0329510
$804$	0	0
$805$	−5.73205	−0.202028
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 8.58682i	− 0.301896i	−0.988542	−	0.150948i	$-0.951767\pi$
$809$	− 8.58682i	− 0.301896i	0.988542	−	0.150948i	$-0.0482327\pi$
$810$	0	0
$811$	27.3397i	0.960028i	0.877261	+	0.480014i	$0.159368\pi$
$811$	27.3397i	0.960028i	−0.877261	+	0.480014i	$0.840632\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	53.5642	1.87627
$816$	0	0
$817$	−23.6603	−0.827768
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 44.4698i	− 1.55201i	−0.630730	−	0.776003i	$-0.717244\pi$
$821$	− 44.4698i	− 1.55201i	0.630730	−	0.776003i	$-0.282756\pi$
$822$	0	0
$823$	20.9282i	0.729511i	0.931103	+	0.364756i	$0.118847\pi$
$823$	20.9282i	0.729511i	−0.931103	+	0.364756i	$0.881153\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−53.2968	−1.85331	−0.926656	−	0.375911i	$-0.877330\pi$
$827$	−53.2968	−1.85331	−0.926656	+	0.375911i	$0.877330\pi$
$828$	0	0
$829$	25.8038	0.896205	0.448102	−	0.893982i	$-0.352100\pi$
$829$	25.8038	0.896205	0.448102	+	0.893982i	$0.352100\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.378937i	0.0131294i
$834$	0	0
$835$	67.9615i	2.35191i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13.0797	−0.451560	−0.225780	−	0.974178i	$-0.572493\pi$
$839$	−13.0797	−0.451560	−0.225780	+	0.974178i	$0.572493\pi$
$840$	0	0
$841$	11.0000	0.379310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 33.8024i	− 1.16284i
$846$	0	0
$847$	11.6603i	0.400651i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.76028	0.163180
$852$	0	0
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.3552i	0.490363i	0.969477	+	0.245182i	$0.0788476\pi$
$857$	14.3552i	0.490363i	−0.969477	+	0.245182i	$0.921152\pi$
$858$	0	0
$859$	− 21.1051i	− 0.720097i	−0.932934	−	0.360049i	$-0.882760\pi$
$859$	− 21.1051i	− 0.720097i	0.932934	−	0.360049i	$-0.117240\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.0512	1.36336	0.681680	−	0.731650i	$-0.261250\pi$
$863$	40.0512	1.36336	0.681680	+	0.731650i	$0.261250\pi$
$864$	0	0
$865$	30.6077	1.04069
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 8.58682i	− 0.291288i
$870$	0	0
$871$	− 9.71281i	− 0.329106i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.54914	−0.119983
$876$	0	0
$877$	−45.8564	−1.54846	−0.774230	−	0.632904i	$-0.781863\pi$
$877$	−45.8564	−1.54846	−0.774230	+	0.632904i	$0.781863\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 43.3973i	− 1.46209i	−0.682328	−	0.731046i	$-0.739032\pi$
$881$	− 43.3973i	− 1.46209i	0.682328	−	0.731046i	$-0.260968\pi$
$882$	0	0
$883$	10.3923i	0.349729i	0.984593	+	0.174864i	$0.0559487\pi$
$883$	10.3923i	0.349729i	−0.984593	+	0.174864i	$0.944051\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.5574	0.858133	0.429066	−	0.903273i	$-0.358843\pi$
$887$	25.5574	0.858133	0.429066	+	0.903273i	$0.358843\pi$
$888$	0	0
$889$	17.6603	0.592306
$890$	0	0
$891$	0	0
$892$	0	0
$893$	16.0368i	0.536652i
$894$	0	0
$895$	35.5167i	1.18719i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−44.3954	−1.48067
$900$	0	0
$901$	−3.32051	−0.110622
$902$	0	0
$903$	0	0
$904$	0	0
$905$	73.5391i	2.44452i
$906$	0	0
$907$	− 15.0718i	− 0.500451i	−0.968188	−	0.250225i	$-0.919495\pi$
$907$	− 15.0718i	− 0.500451i	0.968188	−	0.250225i	$-0.0805047\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42.6023	−1.41148	−0.705738	−	0.708473i	$-0.749384\pi$
$911$	−42.6023	−1.41148	−0.705738	+	0.708473i	$0.749384\pi$
$912$	0	0
$913$	37.2679	1.23339
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.41421i	− 0.0467014i
$918$	0	0
$919$	− 25.7128i	− 0.848187i	−0.905618	−	0.424094i	$-0.860593\pi$
$919$	− 25.7128i	− 0.848187i	0.905618	−	0.424094i	$-0.139407\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.304608	−0.0100263
$924$	0	0
$925$	−9.37307	−0.308185
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40.0512i	1.31404i	0.753874	+	0.657019i	$0.228183\pi$
$929$	40.0512i	1.31404i	−0.753874	+	0.657019i	$0.771817\pi$
$930$	0	0
$931$	− 5.00000i	− 0.163868i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.35225	−0.175037
$936$	0	0
$937$	31.5692	1.03132	0.515661	−	0.856793i	$-0.327547\pi$
$937$	31.5692	1.03132	0.515661	+	0.856793i	$0.327547\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 6.55343i	− 0.213636i	−0.994279	−	0.106818i	$-0.965934\pi$
$941$	− 6.55343i	− 0.213636i	0.994279	−	0.106818i	$-0.0340662\pi$
$942$	0	0
$943$	− 20.6603i	− 0.672790i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.2765	−0.366438	−0.183219	−	0.983072i	$-0.558652\pi$
$947$	−11.2765	−0.366438	−0.183219	+	0.983072i	$0.558652\pi$
$948$	0	0
$949$	0.248711	0.00807351
$950$	0	0
$951$	0	0
$952$	0	0
$953$	59.5728i	1.92975i	0.262703	+	0.964877i	$0.415386\pi$
$953$	59.5728i	1.92975i	−0.262703	+	0.964877i	$0.584614\pi$
$954$	0	0
$955$	− 14.9474i	− 0.483688i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.48477	0.112529
$960$	0	0
$961$	−78.4974	−2.53217
$962$	0	0
$963$	0	0
$964$	0	0
$965$	27.6551i	0.890250i
$966$	0	0
$967$	− 24.2487i	− 0.779786i	−0.920860	−	0.389893i	$-0.872512\pi$
$967$	− 24.2487i	− 0.779786i	0.920860	−	0.389893i	$-0.127488\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.0368	0.514646	0.257323	−	0.966325i	$-0.417160\pi$
$971$	16.0368	0.514646	0.257323	+	0.966325i	$0.417160\pi$
$972$	0	0
$973$	−11.3205	−0.362919
$974$	0	0
$975$	0	0
$976$	0	0
$977$	41.2624i	1.32010i	0.751221	+	0.660050i	$0.229465\pi$
$977$	41.2624i	1.32010i	−0.751221	+	0.660050i	$0.770535\pi$
$978$	0	0
$979$	− 9.19615i	− 0.293910i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−61.1629	−1.95079	−0.975397	−	0.220456i	$-0.929246\pi$
$983$	−61.1629	−1.95079	−0.975397	+	0.220456i	$0.929246\pi$
$984$	0	0
$985$	59.5692	1.89803
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.14162i	0.290687i
$990$	0	0
$991$	32.8372i	1.04311i	0.853219	+	0.521554i	$0.174647\pi$
$991$	32.8372i	1.04311i	−0.853219	+	0.521554i	$0.825353\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−42.3348	−1.34210
$996$	0	0
$997$	−15.0718	−0.477329	−0.238664	−	0.971102i	$-0.576710\pi$
$997$	−15.0718	−0.477329	−0.238664	+	0.971102i	$0.576710\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.b.2591.6	yes	8
3.2	odd	2	inner	6048.2.h.b.2591.4	yes	8
4.3	odd	2	inner	6048.2.h.b.2591.5	yes	8
12.11	even	2	inner	6048.2.h.b.2591.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.b.2591.3	✓	8	12.11	even	2	inner
6048.2.h.b.2591.4	yes	8	3.2	odd	2	inner
6048.2.h.b.2591.5	yes	8	4.3	odd	2	inner
6048.2.h.b.2591.6	yes	8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.96713i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+2.96713i q^{5} +1.00000i q^{7} -4.76028 q^{11} -1.26795 q^{13} -0.378937i q^{17} +5.00000i q^{19} +1.93185 q^{23} -3.80385 q^{25} +4.24264i q^{29} +10.4641i q^{31} -2.96713 q^{35} +2.46410 q^{37} -10.6945i q^{41} +4.73205i q^{43} +3.20736 q^{47} -1.00000 q^{49} -8.76268i q^{53} -14.1244i q^{55} -3.48477 q^{59} -4.92820 q^{61} -3.76217i q^{65} +7.66025i q^{67} +0.240237 q^{71} -0.196152 q^{73} -4.76028i q^{77} +1.80385i q^{79} -7.82894 q^{83} +1.12436 q^{85} +1.93185i q^{89} -1.26795i q^{91} -14.8356 q^{95} -12.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{13} - 72 q^{25} - 8 q^{37} - 8 q^{49} + 16 q^{61} + 40 q^{73} - 88 q^{85} - 48 q^{97}+O(q^{100}) \) 8 * q - 24 * q^13 - 72 * q^25 - 8 * q^37 - 8 * q^49 + 16 * q^61 + 40 * q^73 - 88 * q^85 - 48 * q^97

Introduction

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