LMFDB - Embedded newform 6048.2.c.d.3025.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.c (of order $2$, degree $1$, not 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:	$16$
Coefficient field:	$\Q(\zeta_{40})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.1
Root			$0.156434 - 0.987688i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.d.3025.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.29757i q^{5} +1.00000 q^{7} +O(q^{10})$	$q-4.29757i q^{5} +1.00000 q^{7} +1.60758i q^{11} -6.02967i q^{13} +3.16228 q^{17} +3.07768i q^{19} -5.95080 q^{23} -13.4691 q^{25} +3.53972i q^{29} +2.08831 q^{31} -4.29757i q^{35} -4.48276i q^{37} -6.69568 q^{41} -12.2811i q^{43} -3.82734 q^{47} +1.00000 q^{49} +8.63711i q^{53} +6.90868 q^{55} -1.55265i q^{59} -3.64886i q^{61} -25.9129 q^{65} +0.772361i q^{67} -7.36501 q^{71} +1.32739 q^{73} +1.60758i q^{77} -12.7401 q^{79} +7.08446i q^{83} -13.5901i q^{85} +9.73307 q^{89} -6.02967i q^{91} +13.2266 q^{95} -10.1803 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{7}+O(q^{10}) $ 16 * q + 16 * q^7	$ 16 q + 16 q^{7} - 32 q^{25} - 40 q^{31} + 16 q^{49} + 72 q^{55} + 24 q^{73} - 24 q^{79} + 16 q^{97}+O(q^{100}) $ 16 * q + 16 * q^7 - 32 * q^25 - 40 * q^31 + 16 * q^49 + 72 * q^55 + 24 * q^73 - 24 * q^79 + 16 * 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
$5$	− 4.29757i	− 1.92193i	−0.276666	−	0.960966i	$-0.589230\pi$
$5$	− 4.29757i	− 1.92193i	0.276666	−	0.960966i	$-0.410770\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.60758i	0.484703i	0.970189	+	0.242351i	$0.0779187\pi$
$11$	1.60758i	0.484703i	−0.970189	+	0.242351i	$0.922081\pi$
$12$	0	0
$13$	− 6.02967i	− 1.67233i	−0.548478	−	0.836165i	$-0.684792\pi$
$13$	− 6.02967i	− 1.67233i	0.548478	−	0.836165i	$-0.315208\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.16228	0.766965	0.383482	−	0.923548i	$-0.374725\pi$
$17$	3.16228	0.766965	0.383482	+	0.923548i	$0.374725\pi$
$18$	0	0
$19$	3.07768i	0.706069i	0.935610	+	0.353035i	$0.114850\pi$
$19$	3.07768i	0.706069i	−0.935610	+	0.353035i	$0.885150\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.95080	−1.24083	−0.620413	−	0.784275i	$-0.713035\pi$
$23$	−5.95080	−1.24083	−0.620413	+	0.784275i	$0.713035\pi$
$24$	0	0
$25$	−13.4691	−2.69382
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.53972i	0.657310i	0.944450	+	0.328655i	$0.106595\pi$
$29$	3.53972i	0.657310i	−0.944450	+	0.328655i	$0.893405\pi$
$30$	0	0
$31$	2.08831	0.375071	0.187535	−	0.982258i	$-0.439950\pi$
$31$	2.08831	0.375071	0.187535	+	0.982258i	$0.439950\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 4.29757i	− 0.726422i
$36$	0	0
$37$	− 4.48276i	− 0.736961i	−0.929636	−	0.368480i	$-0.879878\pi$
$37$	− 4.48276i	− 0.736961i	0.929636	−	0.368480i	$-0.120122\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.69568	−1.04569	−0.522845	−	0.852428i	$-0.675129\pi$
$41$	−6.69568	−1.04569	−0.522845	+	0.852428i	$0.675129\pi$
$42$	0	0
$43$	− 12.2811i	− 1.87284i	−0.350875	−	0.936422i	$-0.614116\pi$
$43$	− 12.2811i	− 1.87284i	0.350875	−	0.936422i	$-0.385884\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.82734	−0.558275	−0.279138	−	0.960251i	$-0.590049\pi$
$47$	−3.82734	−0.558275	−0.279138	+	0.960251i	$0.590049\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.63711i	1.18640i	0.805056	+	0.593199i	$0.202135\pi$
$53$	8.63711i	1.18640i	−0.805056	+	0.593199i	$0.797865\pi$
$54$	0	0
$55$	6.90868	0.931566
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.55265i	− 0.202137i	−0.994879	−	0.101069i	$-0.967774\pi$
$59$	− 1.55265i	− 0.202137i	0.994879	−	0.101069i	$-0.0322262\pi$
$60$	0	0
$61$	− 3.64886i	− 0.467189i	−0.972334	−	0.233594i	$-0.924951\pi$
$61$	− 3.64886i	− 0.467189i	0.972334	−	0.233594i	$-0.0750487\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−25.9129	−3.21411
$66$	0	0
$67$	0.772361i	0.0943590i	0.998886	+	0.0471795i	$0.0150233\pi$
$67$	0.772361i	0.0943590i	−0.998886	+	0.0471795i	$0.984977\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.36501	−0.874066	−0.437033	−	0.899446i	$-0.643971\pi$
$71$	−7.36501	−0.874066	−0.437033	+	0.899446i	$0.643971\pi$
$72$	0	0
$73$	1.32739	0.155359	0.0776797	−	0.996978i	$-0.475249\pi$
$73$	1.32739	0.155359	0.0776797	+	0.996978i	$0.475249\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.60758i	0.183200i
$78$	0	0
$79$	−12.7401	−1.43337	−0.716685	−	0.697397i	$-0.754341\pi$
$79$	−12.7401	−1.43337	−0.716685	+	0.697397i	$0.754341\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.08446i	0.777620i	0.921318	+	0.388810i	$0.127114\pi$
$83$	7.08446i	0.777620i	−0.921318	+	0.388810i	$0.872886\pi$
$84$	0	0
$85$	− 13.5901i	− 1.47405i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.73307	1.03170	0.515852	−	0.856678i	$-0.327475\pi$
$89$	9.73307	1.03170	0.515852	+	0.856678i	$0.327475\pi$
$90$	0	0
$91$	− 6.02967i	− 0.632081i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	13.2266	1.35702
$96$	0	0
$97$	−10.1803	−1.03366	−0.516828	−	0.856089i	$-0.672887\pi$
$97$	−10.1803	−1.03366	−0.516828	+	0.856089i	$0.672887\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.02749i	− 0.102239i	−0.998693	−	0.0511194i	$-0.983721\pi$
$101$	− 1.02749i	− 0.102239i	0.998693	−	0.0511194i	$-0.0162789\pi$
$102$	0	0
$103$	0.965118	0.0950959	0.0475479	−	0.998869i	$-0.484859\pi$
$103$	0.965118	0.0950959	0.0475479	+	0.998869i	$0.484859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 16.0949i	− 1.55595i	−0.628294	−	0.777976i	$-0.716246\pi$
$107$	− 16.0949i	− 1.55595i	0.628294	−	0.777976i	$-0.283754\pi$
$108$	0	0
$109$	− 20.2686i	− 1.94138i	−0.240327	−	0.970692i	$-0.577255\pi$
$109$	− 20.2686i	− 1.94138i	0.240327	−	0.970692i	$-0.422745\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.951214	−0.0894826	−0.0447413	−	0.998999i	$-0.514246\pi$
$113$	−0.951214	−0.0894826	−0.0447413	+	0.998999i	$0.514246\pi$
$114$	0	0
$115$	25.5740i	2.38478i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.16228	0.289886
$120$	0	0
$121$	8.41570	0.765063
$122$	0	0
$123$	0	0
$124$	0	0
$125$	36.3966i	3.25541i
$126$	0	0
$127$	11.6132	1.03050	0.515250	−	0.857040i	$-0.327699\pi$
$127$	11.6132	1.03050	0.515250	+	0.857040i	$0.327699\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.0120i	0.874752i	0.899279	+	0.437376i	$0.144092\pi$
$131$	10.0120i	0.874752i	−0.899279	+	0.437376i	$0.855908\pi$
$132$	0	0
$133$	3.07768i	0.266869i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.40383	−0.717988	−0.358994	−	0.933340i	$-0.616880\pi$
$137$	−8.40383	−0.717988	−0.358994	+	0.933340i	$0.616880\pi$
$138$	0	0
$139$	− 6.61437i	− 0.561024i	−0.959851	−	0.280512i	$-0.909496\pi$
$139$	− 6.61437i	− 0.561024i	0.959851	−	0.280512i	$-0.0905042\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	9.69316	0.810583
$144$	0	0
$145$	15.2122	1.26330
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.9866i	1.39160i	0.718238	+	0.695798i	$0.244949\pi$
$149$	16.9866i	1.39160i	−0.718238	+	0.695798i	$0.755051\pi$
$150$	0	0
$151$	11.5256	0.937937	0.468968	−	0.883215i	$-0.344626\pi$
$151$	11.5256	0.937937	0.468968	+	0.883215i	$0.344626\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.97464i	− 0.720860i
$156$	0	0
$157$	10.3889i	0.829127i	0.910020	+	0.414563i	$0.136066\pi$
$157$	10.3889i	0.829127i	−0.910020	+	0.414563i	$0.863934\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.95080	−0.468988
$162$	0	0
$163$	11.8743i	0.930067i	0.885293	+	0.465034i	$0.153958\pi$
$163$	11.8743i	0.930067i	−0.885293	+	0.465034i	$0.846042\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.0596	−1.16535	−0.582673	−	0.812706i	$-0.697993\pi$
$167$	−15.0596	−1.16535	−0.582673	+	0.812706i	$0.697993\pi$
$168$	0	0
$169$	−23.3570	−1.79669
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 23.7854i	− 1.80837i	−0.427142	−	0.904185i	$-0.640479\pi$
$173$	− 23.7854i	− 1.80837i	0.427142	−	0.904185i	$-0.359521\pi$
$174$	0	0
$175$	−13.4691	−1.01817
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.1919i	1.65870i	0.558728	+	0.829351i	$0.311290\pi$
$179$	22.1919i	1.65870i	−0.558728	+	0.829351i	$0.688710\pi$
$180$	0	0
$181$	− 8.60253i	− 0.639421i	−0.947515	−	0.319711i	$-0.896414\pi$
$181$	− 8.60253i	− 0.639421i	0.947515	−	0.319711i	$-0.103586\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−19.2650	−1.41639
$186$	0	0
$187$	5.08361i	0.371750i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.72989	0.631673	0.315836	−	0.948814i	$-0.397715\pi$
$191$	8.72989	0.631673	0.315836	+	0.948814i	$0.397715\pi$
$192$	0	0
$193$	−12.8708	−0.926459	−0.463229	−	0.886238i	$-0.653309\pi$
$193$	−12.8708	−0.926459	−0.463229	+	0.886238i	$0.653309\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.4345i	− 1.59839i	−0.601072	−	0.799195i	$-0.705259\pi$
$197$	− 22.4345i	− 1.59839i	0.601072	−	0.799195i	$-0.294741\pi$
$198$	0	0
$199$	14.9799	1.06189	0.530947	−	0.847405i	$-0.321836\pi$
$199$	14.9799	1.06189	0.530947	+	0.847405i	$0.321836\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.53972i	0.248440i
$204$	0	0
$205$	28.7752i	2.00974i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.94761	−0.342234
$210$	0	0
$211$	8.50651i	0.585612i	0.956172	+	0.292806i	$0.0945890\pi$
$211$	8.50651i	0.585612i	−0.956172	+	0.292806i	$0.905411\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−52.7787	−3.59948
$216$	0	0
$217$	2.08831	0.141763
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 19.0675i	− 1.28262i
$222$	0	0
$223$	−16.3214	−1.09296	−0.546479	−	0.837473i	$-0.684032\pi$
$223$	−16.3214	−1.09296	−0.546479	+	0.837473i	$0.684032\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.8279i	− 0.718671i	−0.933208	−	0.359336i	$-0.883003\pi$
$227$	− 10.8279i	− 0.718671i	0.933208	−	0.359336i	$-0.116997\pi$
$228$	0	0
$229$	24.9251i	1.64710i	0.567245	+	0.823549i	$0.308009\pi$
$229$	24.9251i	1.64710i	−0.567245	+	0.823549i	$0.691991\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.5950	−0.890641	−0.445321	−	0.895371i	$-0.646910\pi$
$233$	−13.5950	−0.890641	−0.445321	+	0.895371i	$0.646910\pi$
$234$	0	0
$235$	16.4483i	1.07297i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.82998	−0.247741	−0.123870	−	0.992298i	$-0.539531\pi$
$239$	−3.82998	−0.247741	−0.123870	+	0.992298i	$0.539531\pi$
$240$	0	0
$241$	5.32509	0.343019	0.171509	−	0.985182i	$-0.445136\pi$
$241$	5.32509	0.343019	0.171509	+	0.985182i	$0.445136\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.29757i	− 0.274562i
$246$	0	0
$247$	18.5574	1.18078
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.2193i	1.21311i	0.795040	+	0.606556i	$0.207450\pi$
$251$	19.2193i	1.21311i	−0.795040	+	0.606556i	$0.792550\pi$
$252$	0	0
$253$	− 9.56636i	− 0.601432i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.655642	0.0408979	0.0204489	−	0.999791i	$-0.493490\pi$
$257$	0.655642	0.0408979	0.0204489	+	0.999791i	$0.493490\pi$
$258$	0	0
$259$	− 4.48276i	− 0.278545i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.8211	−1.59220	−0.796098	−	0.605168i	$-0.793106\pi$
$263$	−25.8211	−1.59220	−0.796098	+	0.605168i	$0.793106\pi$
$264$	0	0
$265$	37.1186	2.28018
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.2078i	0.866266i	0.901330	+	0.433133i	$0.142592\pi$
$269$	14.2078i	0.866266i	−0.901330	+	0.433133i	$0.857408\pi$
$270$	0	0
$271$	−3.52786	−0.214302	−0.107151	−	0.994243i	$-0.534173\pi$
$271$	−3.52786	−0.214302	−0.107151	+	0.994243i	$0.534173\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 21.6526i	− 1.30570i
$276$	0	0
$277$	− 13.9396i	− 0.837548i	−0.908091	−	0.418774i	$-0.862460\pi$
$277$	− 13.9396i	− 0.837548i	0.908091	−	0.418774i	$-0.137540\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.4105	1.21759	0.608793	−	0.793329i	$-0.291654\pi$
$281$	20.4105	1.21759	0.608793	+	0.793329i	$0.291654\pi$
$282$	0	0
$283$	29.6867i	1.76469i	0.470600	+	0.882347i	$0.344037\pi$
$283$	29.6867i	1.76469i	−0.470600	+	0.882347i	$0.655963\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.69568	−0.395233
$288$	0	0
$289$	−7.00000	−0.411765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 16.4504i	− 0.961044i	−0.876983	−	0.480522i	$-0.840447\pi$
$293$	− 16.4504i	− 0.961044i	0.876983	−	0.480522i	$-0.159553\pi$
$294$	0	0
$295$	−6.67261	−0.388494
$296$	0	0
$297$	0	0
$298$	0	0
$299$	35.8813i	2.07507i
$300$	0	0
$301$	− 12.2811i	− 0.707869i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.6812	−0.897905
$306$	0	0
$307$	− 19.8393i	− 1.13229i	−0.824306	−	0.566145i	$-0.808434\pi$
$307$	− 19.8393i	− 1.13229i	0.824306	−	0.566145i	$-0.191566\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−32.3919	−1.83677	−0.918387	−	0.395683i	$-0.870508\pi$
$311$	−32.3919	−1.83677	−0.918387	+	0.395683i	$0.870508\pi$
$312$	0	0
$313$	−6.02156	−0.340359	−0.170179	−	0.985413i	$-0.554435\pi$
$313$	−6.02156	−0.340359	−0.170179	+	0.985413i	$0.554435\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 5.23318i	− 0.293925i	−0.989142	−	0.146962i	$-0.953050\pi$
$317$	− 5.23318i	− 0.293925i	0.989142	−	0.146962i	$-0.0469496\pi$
$318$	0	0
$319$	−5.69038	−0.318600
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.73249i	0.541530i
$324$	0	0
$325$	81.2144i	4.50496i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.82734	−0.211008
$330$	0	0
$331$	16.5740i	0.910988i	0.890239	+	0.455494i	$0.150537\pi$
$331$	16.5740i	0.910988i	−0.890239	+	0.455494i	$0.849463\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.31928	0.181352
$336$	0	0
$337$	1.70020	0.0926159	0.0463080	−	0.998927i	$-0.485254\pi$
$337$	1.70020	0.0926159	0.0463080	+	0.998927i	$0.485254\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.35711i	0.181798i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 24.0032i	− 1.28856i	−0.764790	−	0.644279i	$-0.777157\pi$
$347$	− 24.0032i	− 1.28856i	0.764790	−	0.644279i	$-0.222843\pi$
$348$	0	0
$349$	21.3679i	1.14380i	0.820324	+	0.571898i	$0.193793\pi$
$349$	21.3679i	1.14380i	−0.820324	+	0.571898i	$0.806207\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.1560	1.33892	0.669460	−	0.742849i	$-0.266526\pi$
$353$	25.1560	1.33892	0.669460	+	0.742849i	$0.266526\pi$
$354$	0	0
$355$	31.6517i	1.67990i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.1318	0.956957	0.478479	−	0.878099i	$-0.341188\pi$
$359$	18.1318	0.956957	0.478479	+	0.878099i	$0.341188\pi$
$360$	0	0
$361$	9.52786	0.501467
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 5.70456i	− 0.298590i
$366$	0	0
$367$	29.6249	1.54641	0.773203	−	0.634158i	$-0.218653\pi$
$367$	29.6249	1.54641	0.773203	+	0.634158i	$0.218653\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.63711i	0.448416i
$372$	0	0
$373$	− 29.3696i	− 1.52070i	−0.649514	−	0.760349i	$-0.725028\pi$
$373$	− 29.3696i	− 1.52070i	0.649514	−	0.760349i	$-0.274972\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	21.3434	1.09924
$378$	0	0
$379$	− 16.1375i	− 0.828930i	−0.910065	−	0.414465i	$-0.863969\pi$
$379$	− 16.1375i	− 0.828930i	0.910065	−	0.414465i	$-0.136031\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.9227	−1.12020	−0.560099	−	0.828426i	$-0.689237\pi$
$383$	−21.9227	−1.12020	−0.560099	+	0.828426i	$0.689237\pi$
$384$	0	0
$385$	6.90868	0.352099
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.7725i	1.40812i	0.710140	+	0.704061i	$0.248632\pi$
$389$	27.7725i	1.40812i	−0.710140	+	0.704061i	$0.751368\pi$
$390$	0	0
$391$	−18.8181	−0.951671
$392$	0	0
$393$	0	0
$394$	0	0
$395$	54.7514i	2.75484i
$396$	0	0
$397$	− 27.6193i	− 1.38617i	−0.720855	−	0.693086i	$-0.756251\pi$
$397$	− 27.6193i	− 1.38617i	0.720855	−	0.693086i	$-0.243749\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.4439	0.521546	0.260773	−	0.965400i	$-0.416023\pi$
$401$	10.4439	0.521546	0.260773	+	0.965400i	$0.416023\pi$
$402$	0	0
$403$	− 12.5918i	− 0.627242i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.20638	0.357207
$408$	0	0
$409$	15.1424	0.748745	0.374373	−	0.927278i	$-0.377858\pi$
$409$	15.1424	0.748745	0.374373	+	0.927278i	$0.377858\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.55265i	− 0.0764007i
$414$	0	0
$415$	30.4460	1.49453
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 19.9623i	− 0.975222i	−0.873061	−	0.487611i	$-0.837869\pi$
$419$	− 19.9623i	− 0.975222i	0.873061	−	0.487611i	$-0.162131\pi$
$420$	0	0
$421$	− 12.8339i	− 0.625486i	−0.949838	−	0.312743i	$-0.898752\pi$
$421$	− 12.8339i	− 0.625486i	0.949838	−	0.312743i	$-0.101248\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−42.5931	−2.06607
$426$	0	0
$427$	− 3.64886i	− 0.176581i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.66364	0.128303	0.0641514	−	0.997940i	$-0.479566\pi$
$431$	2.66364	0.128303	0.0641514	+	0.997940i	$0.479566\pi$
$432$	0	0
$433$	−19.2122	−0.923280	−0.461640	−	0.887067i	$-0.652739\pi$
$433$	−19.2122	−0.923280	−0.461640	+	0.887067i	$0.652739\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 18.3147i	− 0.876109i
$438$	0	0
$439$	35.1209	1.67623	0.838114	−	0.545495i	$-0.183658\pi$
$439$	35.1209	1.67623	0.838114	+	0.545495i	$0.183658\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.40943i	0.399544i	0.979842	+	0.199772i	$0.0640202\pi$
$443$	8.40943i	0.399544i	−0.979842	+	0.199772i	$0.935980\pi$
$444$	0	0
$445$	− 41.8286i	− 1.98286i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.81067	−0.227030	−0.113515	−	0.993536i	$-0.536211\pi$
$449$	−4.81067	−0.227030	−0.113515	+	0.993536i	$0.536211\pi$
$450$	0	0
$451$	− 10.7638i	− 0.506848i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−25.9129	−1.21482
$456$	0	0
$457$	−37.8291	−1.76957	−0.884785	−	0.465999i	$-0.845695\pi$
$457$	−37.8291	−1.76957	−0.884785	+	0.465999i	$0.845695\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 10.5113i	− 0.489558i	−0.969579	−	0.244779i	$-0.921285\pi$
$461$	− 10.5113i	− 0.489558i	0.969579	−	0.244779i	$-0.0787154\pi$
$462$	0	0
$463$	−4.76163	−0.221292	−0.110646	−	0.993860i	$-0.535292\pi$
$463$	−4.76163	−0.221292	−0.110646	+	0.993860i	$0.535292\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.7216i	0.912609i	0.889824	+	0.456305i	$0.150827\pi$
$467$	19.7216i	0.912609i	−0.889824	+	0.456305i	$0.849173\pi$
$468$	0	0
$469$	0.772361i	0.0356643i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.7428	0.907773
$474$	0	0
$475$	− 41.4537i	− 1.90203i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.49558	−0.114026	−0.0570131	−	0.998373i	$-0.518158\pi$
$479$	−2.49558	−0.114026	−0.0570131	+	0.998373i	$0.518158\pi$
$480$	0	0
$481$	−27.0296	−1.23244
$482$	0	0
$483$	0	0
$484$	0	0
$485$	43.7507i	1.98662i
$486$	0	0
$487$	10.0178	0.453951	0.226976	−	0.973900i	$-0.427116\pi$
$487$	10.0178	0.453951	0.226976	+	0.973900i	$0.427116\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4.23469i	− 0.191109i	−0.995424	−	0.0955544i	$-0.969538\pi$
$491$	− 4.23469i	− 0.191109i	0.995424	−	0.0955544i	$-0.0304624\pi$
$492$	0	0
$493$	11.1936i	0.504134i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.36501	−0.330366
$498$	0	0
$499$	10.1176i	0.452925i	0.974020	+	0.226463i	$0.0727161\pi$
$499$	10.1176i	0.452925i	−0.974020	+	0.226463i	$0.927284\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.6693	−0.966186	−0.483093	−	0.875569i	$-0.660487\pi$
$503$	−21.6693	−0.966186	−0.483093	+	0.875569i	$0.660487\pi$
$504$	0	0
$505$	−4.41570	−0.196496
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.8362i	0.613280i	0.951826	+	0.306640i	$0.0992048\pi$
$509$	13.8362i	0.613280i	−0.951826	+	0.306640i	$0.900795\pi$
$510$	0	0
$511$	1.32739	0.0587203
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.14766i	− 0.182768i
$516$	0	0
$517$	− 6.15275i	− 0.270597i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.5067	−0.854603	−0.427302	−	0.904109i	$-0.640536\pi$
$521$	−19.5067	−0.854603	−0.427302	+	0.904109i	$0.640536\pi$
$522$	0	0
$523$	− 0.616566i	− 0.0269606i	−0.999909	−	0.0134803i	$-0.995709\pi$
$523$	− 0.616566i	− 0.0269606i	0.999909	−	0.0134803i	$-0.00429104\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.60380	0.287666
$528$	0	0
$529$	12.4120	0.539651
$530$	0	0
$531$	0	0
$532$	0	0
$533$	40.3727i	1.74874i
$534$	0	0
$535$	−69.1690	−2.99044
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.60758i	0.0692433i
$540$	0	0
$541$	33.0638i	1.42152i	0.703432	+	0.710762i	$0.251650\pi$
$541$	33.0638i	1.42152i	−0.703432	+	0.710762i	$0.748350\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−87.1059	−3.73121
$546$	0	0
$547$	− 45.6463i	− 1.95170i	−0.218450	−	0.975848i	$-0.570100\pi$
$547$	− 45.6463i	− 1.95170i	0.218450	−	0.975848i	$-0.429900\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.8941	−0.464106
$552$	0	0
$553$	−12.7401	−0.541763
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.0518694i	0.00219778i	0.999999	+	0.00109889i	$0.000349787\pi$
$557$	0.0518694i	0.00219778i	−0.999999	+	0.00109889i	$0.999650\pi$
$558$	0	0
$559$	−74.0508	−3.13201
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.9105i	0.459822i	0.973212	+	0.229911i	$0.0738436\pi$
$563$	10.9105i	0.459822i	−0.973212	+	0.229911i	$0.926156\pi$
$564$	0	0
$565$	4.08791i	0.171980i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.2746	−1.10149	−0.550745	−	0.834673i	$-0.685656\pi$
$569$	−26.2746	−1.10149	−0.550745	+	0.834673i	$0.685656\pi$
$570$	0	0
$571$	− 24.4590i	− 1.02358i	−0.859111	−	0.511789i	$-0.828983\pi$
$571$	− 24.4590i	− 1.02358i	0.859111	−	0.511789i	$-0.171017\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	80.1520	3.34257
$576$	0	0
$577$	−10.2538	−0.426873	−0.213436	−	0.976957i	$-0.568466\pi$
$577$	−10.2538	−0.426873	−0.213436	+	0.976957i	$0.568466\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.08446i	0.293913i
$582$	0	0
$583$	−13.8848	−0.575050
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 25.3720i	− 1.04722i	−0.851959	−	0.523608i	$-0.824586\pi$
$587$	− 25.3720i	− 1.04722i	0.851959	−	0.523608i	$-0.175414\pi$
$588$	0	0
$589$	6.42714i	0.264826i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.14310	0.293332	0.146666	−	0.989186i	$-0.453146\pi$
$593$	7.14310	0.293332	0.146666	+	0.989186i	$0.453146\pi$
$594$	0	0
$595$	− 13.5901i	− 0.557140i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.00175	−0.122648	−0.0613242	−	0.998118i	$-0.519532\pi$
$599$	−3.00175	−0.122648	−0.0613242	+	0.998118i	$0.519532\pi$
$600$	0	0
$601$	21.2516	0.866871	0.433435	−	0.901185i	$-0.357301\pi$
$601$	21.2516	0.866871	0.433435	+	0.901185i	$0.357301\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 36.1671i	− 1.47040i
$606$	0	0
$607$	−24.0614	−0.976623	−0.488312	−	0.872669i	$-0.662387\pi$
$607$	−24.0614	−0.976623	−0.488312	+	0.872669i	$0.662387\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	23.0776i	0.933620i
$612$	0	0
$613$	− 20.9197i	− 0.844939i	−0.906377	−	0.422469i	$-0.861163\pi$
$613$	− 20.9197i	− 0.844939i	0.906377	−	0.422469i	$-0.138837\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.95122	0.159070	0.0795350	−	0.996832i	$-0.474656\pi$
$617$	3.95122	0.159070	0.0795350	+	0.996832i	$0.474656\pi$
$618$	0	0
$619$	− 23.8651i	− 0.959220i	−0.877482	−	0.479610i	$-0.840778\pi$
$619$	− 23.8651i	− 0.959220i	0.877482	−	0.479610i	$-0.159222\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.73307	0.389947
$624$	0	0
$625$	89.0716	3.56286
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 14.1757i	− 0.565223i
$630$	0	0
$631$	−36.3725	−1.44797	−0.723983	−	0.689818i	$-0.757690\pi$
$631$	−36.3725	−1.44797	−0.723983	+	0.689818i	$0.757690\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 49.9083i	− 1.98055i
$636$	0	0
$637$	− 6.02967i	− 0.238904i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.876669	0.0346264	0.0173132	−	0.999850i	$-0.494489\pi$
$641$	0.876669	0.0346264	0.0173132	+	0.999850i	$0.494489\pi$
$642$	0	0
$643$	− 15.2486i	− 0.601347i	−0.953727	−	0.300674i	$-0.902789\pi$
$643$	− 15.2486i	− 0.601347i	0.953727	−	0.300674i	$-0.0972114\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.53027	0.374673	0.187337	−	0.982296i	$-0.440014\pi$
$647$	9.53027	0.374673	0.187337	+	0.982296i	$0.440014\pi$
$648$	0	0
$649$	2.49600	0.0979765
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.29496i	0.363740i	0.983323	+	0.181870i	$0.0582150\pi$
$653$	9.29496i	0.363740i	−0.983323	+	0.181870i	$0.941785\pi$
$654$	0	0
$655$	43.0273	1.68121
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 36.1879i	− 1.40968i	−0.709366	−	0.704841i	$-0.751018\pi$
$659$	− 36.1879i	− 1.40968i	0.709366	−	0.704841i	$-0.248982\pi$
$660$	0	0
$661$	19.8091i	0.770483i	0.922816	+	0.385242i	$0.125882\pi$
$661$	19.8091i	0.770483i	−0.922816	+	0.385242i	$0.874118\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13.2266	0.512904
$666$	0	0
$667$	− 21.0642i	− 0.815607i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	5.86582	0.226448
$672$	0	0
$673$	−0.186375	−0.00718421	−0.00359211	−	0.999994i	$-0.501143\pi$
$673$	−0.186375	−0.00718421	−0.00359211	+	0.999994i	$0.501143\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.31885i	0.127554i	0.997964	+	0.0637770i	$0.0203146\pi$
$677$	3.31885i	0.127554i	−0.997964	+	0.0637770i	$0.979685\pi$
$678$	0	0
$679$	−10.1803	−0.390686
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 17.9992i	− 0.688720i	−0.938838	−	0.344360i	$-0.888096\pi$
$683$	− 17.9992i	− 0.688720i	0.938838	−	0.344360i	$-0.111904\pi$
$684$	0	0
$685$	36.1161i	1.37992i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	52.0789	1.98405
$690$	0	0
$691$	− 3.67153i	− 0.139671i	−0.997559	−	0.0698357i	$-0.977752\pi$
$691$	− 3.67153i	− 0.139671i	0.997559	−	0.0698357i	$-0.0222475\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.4257	−1.07825
$696$	0	0
$697$	−21.1736	−0.802007
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.06843i	0.266971i	0.991051	+	0.133485i	$0.0426169\pi$
$701$	7.06843i	0.266971i	−0.991051	+	0.133485i	$0.957383\pi$
$702$	0	0
$703$	13.7965	0.520345
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.02749i	− 0.0386426i
$708$	0	0
$709$	− 20.4607i	− 0.768417i	−0.923246	−	0.384209i	$-0.874474\pi$
$709$	− 20.4607i	− 0.768417i	0.923246	−	0.384209i	$-0.125526\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.4271	−0.465398
$714$	0	0
$715$	− 41.6571i	− 1.55789i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.5191	1.51111	0.755554	−	0.655087i	$-0.227368\pi$
$719$	40.5191	1.51111	0.755554	+	0.655087i	$0.227368\pi$
$720$	0	0
$721$	0.965118	0.0359429
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 47.6769i	− 1.77068i
$726$	0	0
$727$	−6.87078	−0.254823	−0.127412	−	0.991850i	$-0.540667\pi$
$727$	−6.87078	−0.254823	−0.127412	+	0.991850i	$0.540667\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 38.8361i	− 1.43641i
$732$	0	0
$733$	− 18.5847i	− 0.686442i	−0.939255	−	0.343221i	$-0.888482\pi$
$733$	− 18.5847i	− 0.686442i	0.939255	−	0.343221i	$-0.111518\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.24163	−0.0457360
$738$	0	0
$739$	4.69519i	0.172715i	0.996264	+	0.0863576i	$0.0275228\pi$
$739$	4.69519i	0.172715i	−0.996264	+	0.0863576i	$0.972477\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.5867	0.571821	0.285911	−	0.958256i	$-0.407704\pi$
$743$	15.5867	0.571821	0.285911	+	0.958256i	$0.407704\pi$
$744$	0	0
$745$	73.0011	2.67455
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 16.0949i	− 0.588095i
$750$	0	0
$751$	18.7401	0.683835	0.341917	−	0.939730i	$-0.388924\pi$
$751$	18.7401	0.683835	0.341917	+	0.939730i	$0.388924\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 49.5319i	− 1.80265i
$756$	0	0
$757$	− 24.9095i	− 0.905352i	−0.891675	−	0.452676i	$-0.850469\pi$
$757$	− 24.9095i	− 0.905352i	0.891675	−	0.452676i	$-0.149531\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.70991	0.170734	0.0853670	−	0.996350i	$-0.472794\pi$
$761$	4.70991	0.170734	0.0853670	+	0.996350i	$0.472794\pi$
$762$	0	0
$763$	− 20.2686i	− 0.733774i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.36195	−0.338040
$768$	0	0
$769$	13.8989	0.501208	0.250604	−	0.968090i	$-0.419371\pi$
$769$	13.8989	0.501208	0.250604	+	0.968090i	$0.419371\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 14.8995i	− 0.535899i	−0.963433	−	0.267949i	$-0.913654\pi$
$773$	− 14.8995i	− 0.535899i	0.963433	−	0.267949i	$-0.0863460\pi$
$774$	0	0
$775$	−28.1276	−1.01037
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 20.6072i	− 0.738329i
$780$	0	0
$781$	− 11.8398i	− 0.423662i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	44.6472	1.59353
$786$	0	0
$787$	10.1839i	0.363018i	0.983389	+	0.181509i	$0.0580981\pi$
$787$	10.1839i	0.363018i	−0.983389	+	0.181509i	$0.941902\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.951214	−0.0338213
$792$	0	0
$793$	−22.0014	−0.781293
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14.0210i	− 0.496648i	−0.968677	−	0.248324i	$-0.920120\pi$
$797$	− 14.0210i	− 0.496648i	0.968677	−	0.248324i	$-0.0798798\pi$
$798$	0	0
$799$	−12.1031	−0.428177
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.13388i	0.0753031i
$804$	0	0
$805$	25.5740i	0.901364i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.0399	0.985831	0.492916	−	0.870077i	$-0.335931\pi$
$809$	28.0399	0.985831	0.492916	+	0.870077i	$0.335931\pi$
$810$	0	0
$811$	− 13.2362i	− 0.464785i	−0.972622	−	0.232393i	$-0.925345\pi$
$811$	− 13.2362i	− 0.464785i	0.972622	−	0.232393i	$-0.0746554\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	51.0307	1.78753
$816$	0	0
$817$	37.7972	1.32236
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.8343i	0.482821i	0.970423	+	0.241411i	$0.0776100\pi$
$821$	13.8343i	0.482821i	−0.970423	+	0.241411i	$0.922390\pi$
$822$	0	0
$823$	−29.4862	−1.02782	−0.513912	−	0.857843i	$-0.671804\pi$
$823$	−29.4862	−1.02782	−0.513912	+	0.857843i	$0.671804\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.9797i	1.00772i	0.863784	+	0.503861i	$0.168088\pi$
$827$	28.9797i	1.00772i	−0.863784	+	0.503861i	$0.831912\pi$
$828$	0	0
$829$	− 32.5800i	− 1.13155i	−0.824560	−	0.565774i	$-0.808577\pi$
$829$	− 32.5800i	− 1.13155i	0.824560	−	0.565774i	$-0.191423\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.16228	0.109566
$834$	0	0
$835$	64.7197i	2.23972i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.8215	−0.684313	−0.342157	−	0.939643i	$-0.611157\pi$
$839$	−19.8215	−0.684313	−0.342157	+	0.939643i	$0.611157\pi$
$840$	0	0
$841$	16.4704	0.567944
$842$	0	0
$843$	0	0
$844$	0	0
$845$	100.378i	3.45311i
$846$	0	0
$847$	8.41570	0.289167
$848$	0	0
$849$	0	0
$850$	0	0
$851$	26.6760i	0.914441i
$852$	0	0
$853$	17.4764i	0.598381i	0.954193	+	0.299190i	$0.0967166\pi$
$853$	17.4764i	0.598381i	−0.954193	+	0.299190i	$0.903283\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.0767	0.890764	0.445382	−	0.895341i	$-0.353068\pi$
$857$	26.0767	0.890764	0.445382	+	0.895341i	$0.353068\pi$
$858$	0	0
$859$	− 40.6339i	− 1.38641i	−0.720740	−	0.693205i	$-0.756198\pi$
$859$	− 40.6339i	− 1.38641i	0.720740	−	0.693205i	$-0.243802\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.4216	−1.41001	−0.705004	−	0.709204i	$-0.749055\pi$
$863$	−41.4216	−1.41001	−0.705004	+	0.709204i	$0.749055\pi$
$864$	0	0
$865$	−102.219	−3.47556
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 20.4806i	− 0.694758i
$870$	0	0
$871$	4.65709	0.157799
$872$	0	0
$873$	0	0
$874$	0	0
$875$	36.3966i	1.23043i
$876$	0	0
$877$	33.0093i	1.11464i	0.830296	+	0.557322i	$0.188171\pi$
$877$	33.0093i	1.11464i	−0.830296	+	0.557322i	$0.811829\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18.0587	−0.608414	−0.304207	−	0.952606i	$-0.598391\pi$
$881$	−18.0587	−0.608414	−0.304207	+	0.952606i	$0.598391\pi$
$882$	0	0
$883$	46.6771i	1.57081i	0.618983	+	0.785404i	$0.287545\pi$
$883$	46.6771i	1.57081i	−0.618983	+	0.785404i	$0.712455\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.2001	0.980444	0.490222	−	0.871598i	$-0.336916\pi$
$887$	29.2001	0.980444	0.490222	+	0.871598i	$0.336916\pi$
$888$	0	0
$889$	11.6132	0.389493
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 11.7793i	− 0.394181i
$894$	0	0
$895$	95.3714	3.18791
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.39202i	0.246538i
$900$	0	0
$901$	27.3129i	0.909926i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.9700	−1.22892
$906$	0	0
$907$	− 9.06592i	− 0.301029i	−0.988608	−	0.150514i	$-0.951907\pi$
$907$	− 9.06592i	− 0.301029i	0.988608	−	0.150514i	$-0.0480930\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14.3571	0.475673	0.237837	−	0.971305i	$-0.423562\pi$
$911$	14.3571	0.475673	0.237837	+	0.971305i	$0.423562\pi$
$912$	0	0
$913$	−11.3888	−0.376915
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.0120i	0.330625i
$918$	0	0
$919$	−49.8788	−1.64535	−0.822675	−	0.568513i	$-0.807519\pi$
$919$	−49.8788	−1.64535	−0.822675	+	0.568513i	$0.807519\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	44.4086i	1.46173i
$924$	0	0
$925$	60.3788i	1.98524i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−39.3069	−1.28962	−0.644809	−	0.764343i	$-0.723063\pi$
$929$	−39.3069	−1.28962	−0.644809	+	0.764343i	$0.723063\pi$
$930$	0	0
$931$	3.07768i	0.100867i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	21.8472	0.714478
$936$	0	0
$937$	37.3504	1.22018	0.610092	−	0.792331i	$-0.291133\pi$
$937$	37.3504	1.22018	0.610092	+	0.792331i	$0.291133\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 38.7781i	− 1.26413i	−0.774915	−	0.632065i	$-0.782208\pi$
$941$	− 38.7781i	− 1.26413i	0.774915	−	0.632065i	$-0.217792\pi$
$942$	0	0
$943$	39.8446	1.29752
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.75328i	0.121965i	0.998139	+	0.0609826i	$0.0194234\pi$
$947$	3.75328i	0.121965i	−0.998139	+	0.0609826i	$0.980577\pi$
$948$	0	0
$949$	− 8.00373i	− 0.259812i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	46.5443	1.50772	0.753859	−	0.657036i	$-0.228190\pi$
$953$	46.5443	1.50772	0.753859	+	0.657036i	$0.228190\pi$
$954$	0	0
$955$	− 37.5173i	− 1.21403i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.40383	−0.271374
$960$	0	0
$961$	−26.6390	−0.859322
$962$	0	0
$963$	0	0
$964$	0	0
$965$	55.3131i	1.78059i
$966$	0	0
$967$	16.6487	0.535388	0.267694	−	0.963504i	$-0.413738\pi$
$967$	16.6487	0.535388	0.267694	+	0.963504i	$0.413738\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.74652i	0.184415i	0.995740	+	0.0922073i	$0.0293922\pi$
$971$	5.74652i	0.184415i	−0.995740	+	0.0922073i	$0.970608\pi$
$972$	0	0
$973$	− 6.61437i	− 0.212047i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.6236	0.883758	0.441879	−	0.897075i	$-0.354312\pi$
$977$	27.6236	0.883758	0.441879	+	0.897075i	$0.354312\pi$
$978$	0	0
$979$	15.6467i	0.500070i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.19812	−0.0701091	−0.0350545	−	0.999385i	$-0.511160\pi$
$983$	−2.19812	−0.0701091	−0.0350545	+	0.999385i	$0.511160\pi$
$984$	0	0
$985$	−96.4138	−3.07200
$986$	0	0
$987$	0	0
$988$	0	0
$989$	73.0821i	2.32388i
$990$	0	0
$991$	−2.62346	−0.0833369	−0.0416684	−	0.999131i	$-0.513267\pi$
$991$	−2.62346	−0.0833369	−0.0416684	+	0.999131i	$0.513267\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 64.3770i	− 2.04089i
$996$	0	0
$997$	− 29.1361i	− 0.922749i	−0.887205	−	0.461375i	$-0.847356\pi$
$997$	− 29.1361i	− 0.922749i	0.887205	−	0.461375i	$-0.152644\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.d.3025.1		16
3.2	odd	2	inner	6048.2.c.d.3025.15		16
4.3	odd	2		1512.2.c.d.757.9	yes	16
8.3	odd	2		1512.2.c.d.757.12	yes	16
8.5	even	2	inner	6048.2.c.d.3025.16		16
12.11	even	2		1512.2.c.d.757.8	yes	16
24.5	odd	2	inner	6048.2.c.d.3025.2		16
24.11	even	2		1512.2.c.d.757.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.d.757.5	✓	16	24.11	even	2
1512.2.c.d.757.8	yes	16	12.11	even	2
1512.2.c.d.757.9	yes	16	4.3	odd	2
1512.2.c.d.757.12	yes	16	8.3	odd	2
6048.2.c.d.3025.1		16	1.1	even	1	trivial
6048.2.c.d.3025.2		16	24.5	odd	2	inner
6048.2.c.d.3025.15		16	3.2	odd	2	inner
6048.2.c.d.3025.16		16	8.5	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.29757i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-4.29757i q^{5} +1.00000 q^{7} +1.60758i q^{11} -6.02967i q^{13} +3.16228 q^{17} +3.07768i q^{19} -5.95080 q^{23} -13.4691 q^{25} +3.53972i q^{29} +2.08831 q^{31} -4.29757i q^{35} -4.48276i q^{37} -6.69568 q^{41} -12.2811i q^{43} -3.82734 q^{47} +1.00000 q^{49} +8.63711i q^{53} +6.90868 q^{55} -1.55265i q^{59} -3.64886i q^{61} -25.9129 q^{65} +0.772361i q^{67} -7.36501 q^{71} +1.32739 q^{73} +1.60758i q^{77} -12.7401 q^{79} +7.08446i q^{83} -13.5901i q^{85} +9.73307 q^{89} -6.02967i q^{91} +13.2266 q^{95} -10.1803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{7}+O(q^{10}) \) 16 * q + 16 * q^7	\( 16 q + 16 q^{7} - 32 q^{25} - 40 q^{31} + 16 q^{49} + 72 q^{55} + 24 q^{73} - 24 q^{79} + 16 q^{97}+O(q^{100}) \) 16 * q + 16 * q^7 - 32 * q^25 - 40 * q^31 + 16 * q^49 + 72 * q^55 + 24 * q^73 - 24 * q^79 + 16 * q^97

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