LMFDB - Embedded newform 3024.2.k.d.1889.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3024,2,Mod(1889,3024)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3024.1889"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3024, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,6,0,-4,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3024.1889
Dual form			3024.2.k.d.1889.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} +5.19615i q^{11} -3.46410i q^{13} -6.00000 q^{17} +1.73205i q^{19} -5.19615i q^{23} +4.00000 q^{25} +10.3923i q^{29} +5.19615i q^{31} +(-6.00000 + 5.19615i) q^{35} +1.00000 q^{37} -3.00000 q^{41} -10.0000 q^{43} +6.00000 q^{47} +(1.00000 - 6.92820i) q^{49} +15.5885i q^{55} -6.00000 q^{59} +13.8564i q^{61} -10.3923i q^{65} -2.00000 q^{67} -5.19615i q^{71} +3.46410i q^{73} +(-9.00000 - 10.3923i) q^{77} -14.0000 q^{79} -6.00000 q^{83} -18.0000 q^{85} +9.00000 q^{89} +(6.00000 + 6.92820i) q^{91} +5.19615i q^{95} +6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} - 4 q^{7} - 12 q^{17} + 8 q^{25} - 12 q^{35} + 2 q^{37} - 6 q^{41} - 20 q^{43} + 12 q^{47} + 2 q^{49} - 12 q^{59} - 4 q^{67} - 18 q^{77} - 28 q^{79} - 12 q^{83} - 36 q^{85} + 18 q^{89} + 12 q^{91}+O(q^{100}) $ 2 * q + 6 * q^5 - 4 * q^7 - 12 * q^17 + 8 * q^25 - 12 * q^35 + 2 * q^37 - 6 * q^41 - 20 * q^43 + 12 * q^47 + 2 * q^49 - 12 * q^59 - 4 * q^67 - 18 * q^77 - 28 * q^79 - 12 * q^83 - 36 * q^85 + 18 * q^89 + 12 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	3.00000			1.34164			0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000			1.34164			0.670820	+	0.741620i	$0.265942\pi$
$6$		0			0
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			5.19615i			1.56670i	0.621582	+	0.783349i	$0.286490\pi$
$11$			5.19615i			1.56670i	−0.621582	+	0.783349i	$0.713510\pi$
$12$		0			0
$13$		−	3.46410i		−	0.960769i	−0.877058	−	0.480384i	$-0.840497\pi$
$13$		−	3.46410i		−	0.960769i	0.877058	−	0.480384i	$-0.159503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$			1.73205i			0.397360i	0.980064	+	0.198680i	$0.0636654\pi$
$19$			1.73205i			0.397360i	−0.980064	+	0.198680i	$0.936335\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.19615i		−	1.08347i	−0.840548	−	0.541736i	$-0.817767\pi$
$23$		−	5.19615i		−	1.08347i	0.840548	−	0.541736i	$-0.182233\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			10.3923i			1.92980i	0.262613	+	0.964901i	$0.415416\pi$
$29$			10.3923i			1.92980i	−0.262613	+	0.964901i	$0.584584\pi$
$30$		0			0
$31$			5.19615i			0.933257i	0.884454	+	0.466628i	$0.154531\pi$
$31$			5.19615i			0.933257i	−0.884454	+	0.466628i	$0.845469\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000	+	5.19615i	−1.01419	+	0.878310i
$36$		0			0
$37$	1.00000			0.164399			0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000			0.164399			0.0821995	+	0.996616i	$0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$	−10.0000			−1.52499			−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000			−1.52499			−0.762493	+	0.646997i	$0.776025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			15.5885i			2.10195i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$			13.8564i			1.77413i	0.461644	+	0.887066i	$0.347260\pi$
$61$			13.8564i			1.77413i	−0.461644	+	0.887066i	$0.652740\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	10.3923i		−	1.28901i
$66$		0			0
$67$	−2.00000			−0.244339			−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000			−0.244339			−0.122169	+	0.992509i	$0.538985\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.19615i		−	0.616670i	−0.951278	−	0.308335i	$-0.900228\pi$
$71$		−	5.19615i		−	0.616670i	0.951278	−	0.308335i	$-0.0997717\pi$
$72$		0			0
$73$			3.46410i			0.405442i	0.979236	+	0.202721i	$0.0649785\pi$
$73$			3.46410i			0.405442i	−0.979236	+	0.202721i	$0.935021\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−9.00000	−	10.3923i	−1.02565	−	1.18431i
$78$		0			0
$79$	−14.0000			−1.57512			−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000			−1.57512			−0.787562	+	0.616236i	$0.788657\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	−18.0000			−1.95237
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000			0.953998			0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000			0.953998			0.476999	+	0.878904i	$0.341725\pi$
$90$		0			0
$91$	6.00000	+	6.92820i	0.628971	+	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5.19615i			0.533114i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$		−	8.66025i		−	0.853320i	−0.904412	−	0.426660i	$-0.859690\pi$
$103$		−	8.66025i		−	0.853320i	0.904412	−	0.426660i	$-0.140310\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	10.3923i		−	1.00466i	−0.864675	−	0.502331i	$-0.832476\pi$
$107$		−	10.3923i		−	1.00466i	0.864675	−	0.502331i	$-0.167524\pi$
$108$		0			0
$109$	−11.0000			−1.05361			−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000			−1.05361			−0.526804	+	0.849987i	$0.676610\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	15.5885i		−	1.45363i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.0000	−	10.3923i	1.10004	−	0.952661i
$120$		0			0
$121$	−16.0000			−1.45455
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$	−3.00000	−	3.46410i	−0.260133	−	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.3923i			0.887875i	0.896058	+	0.443937i	$0.146419\pi$
$137$			10.3923i			0.887875i	−0.896058	+	0.443937i	$0.853581\pi$
$138$		0			0
$139$			17.3205i			1.46911i	0.678551	+	0.734553i	$0.262608\pi$
$139$			17.3205i			1.46911i	−0.678551	+	0.734553i	$0.737392\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	18.0000			1.50524
$144$		0			0
$145$			31.1769i			2.58910i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			15.5885i			1.25210i
$156$		0			0
$157$			10.3923i			0.829396i	0.909959	+	0.414698i	$0.136113\pi$
$157$			10.3923i			0.829396i	−0.909959	+	0.414698i	$0.863887\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.00000	+	10.3923i	0.709299	+	0.819028i
$162$		0			0
$163$	10.0000			0.783260			0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000			0.783260			0.391630	+	0.920123i	$0.371911\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000			−0.928588			−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000			−0.928588			−0.464294	+	0.885681i	$0.653692\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$		0			0
$175$	−8.00000	+	6.92820i	−0.604743	+	0.523723i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			10.3923i			0.776757i	0.921500	+	0.388379i	$0.126965\pi$
$179$			10.3923i			0.776757i	−0.921500	+	0.388379i	$0.873035\pi$
$180$		0			0
$181$			6.92820i			0.514969i	0.966282	+	0.257485i	$0.0828937\pi$
$181$			6.92820i			0.514969i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000			0.220564
$186$		0			0
$187$		−	31.1769i		−	2.27988i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	5.19615i		−	0.375980i	−0.982171	−	0.187990i	$-0.939803\pi$
$191$		−	5.19615i		−	0.375980i	0.982171	−	0.187990i	$-0.0601973\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	20.7846i		−	1.48084i	−0.672143	−	0.740421i	$-0.734626\pi$
$197$		−	20.7846i		−	1.48084i	0.672143	−	0.740421i	$-0.265374\pi$
$198$		0			0
$199$		−	1.73205i		−	0.122782i	−0.998114	−	0.0613909i	$-0.980446\pi$
$199$		−	1.73205i		−	0.122782i	0.998114	−	0.0613909i	$-0.0195536\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−18.0000	−	20.7846i	−1.26335	−	1.45879i
$204$		0			0
$205$	−9.00000			−0.628587
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.00000			−0.622543
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−30.0000			−2.04598
$216$		0			0
$217$	−9.00000	−	10.3923i	−0.610960	−	0.705476i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			20.7846i			1.39812i
$222$		0			0
$223$			25.9808i			1.73980i	0.493228	+	0.869900i	$0.335817\pi$
$223$			25.9808i			1.73980i	−0.493228	+	0.869900i	$0.664183\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$			20.7846i			1.37349i	0.726900	+	0.686743i	$0.240960\pi$
$229$			20.7846i			1.37349i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.7846i		−	1.36165i	−0.732448	−	0.680823i	$-0.761622\pi$
$233$		−	20.7846i		−	1.36165i	0.732448	−	0.680823i	$-0.238378\pi$
$234$		0			0
$235$	18.0000			1.17419
$236$		0			0
$237$		0			0
$238$		0			0
$239$			10.3923i			0.672222i	0.941822	+	0.336111i	$0.109112\pi$
$239$			10.3923i			0.672222i	−0.941822	+	0.336111i	$0.890888\pi$
$240$		0			0
$241$			3.46410i			0.223142i	0.993756	+	0.111571i	$0.0355883\pi$
$241$			3.46410i			0.223142i	−0.993756	+	0.111571i	$0.964412\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.00000	−	20.7846i	0.191663	−	1.32788i
$246$		0			0
$247$	6.00000			0.381771
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$	27.0000			1.69748
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000			0.561405			0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000			0.561405			0.280702	+	0.959795i	$0.409433\pi$
$258$		0			0
$259$	−2.00000	+	1.73205i	−0.124274	+	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5.19615i		−	0.320408i	−0.987084	−	0.160204i	$-0.948785\pi$
$263$		−	5.19615i		−	0.320408i	0.987084	−	0.160204i	$-0.0512153\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.0000			−1.28039			−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000			−1.28039			−0.640196	+	0.768211i	$0.721147\pi$
$270$		0			0
$271$			3.46410i			0.210429i	0.994450	+	0.105215i	$0.0335529\pi$
$271$			3.46410i			0.210429i	−0.994450	+	0.105215i	$0.966447\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			20.7846i			1.25336i
$276$		0			0
$277$	−19.0000			−1.14160			−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000			−1.14160			−0.570800	+	0.821089i	$0.693367\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	10.3923i		−	0.619953i	−0.950744	−	0.309976i	$-0.899679\pi$
$281$		−	10.3923i		−	0.619953i	0.950744	−	0.309976i	$-0.100321\pi$
$282$		0			0
$283$		−	10.3923i		−	0.617758i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$		−	10.3923i		−	0.617758i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.00000	−	5.19615i	0.354169	−	0.306719i
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	−18.0000			−1.04800
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−18.0000			−1.04097
$300$		0			0
$301$	20.0000	−	17.3205i	1.15278	−	0.998337i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			41.5692i			2.38025i
$306$		0			0
$307$			15.5885i			0.889680i	0.895610	+	0.444840i	$0.146740\pi$
$307$			15.5885i			0.889680i	−0.895610	+	0.444840i	$0.853260\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.00000			0.340229			0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000			0.340229			0.170114	+	0.985424i	$0.445586\pi$
$312$		0			0
$313$		−	3.46410i		−	0.195803i	−0.995196	−	0.0979013i	$-0.968787\pi$
$313$		−	3.46410i		−	0.195803i	0.995196	−	0.0979013i	$-0.0312129\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			31.1769i			1.75107i	0.483155	+	0.875535i	$0.339491\pi$
$317$			31.1769i			1.75107i	−0.483155	+	0.875535i	$0.660509\pi$
$318$		0			0
$319$	−54.0000			−3.02342
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	10.3923i		−	0.578243i
$324$		0			0
$325$		−	13.8564i		−	0.768615i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000	+	10.3923i	−0.661581	+	0.572946i
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.00000			−0.327815
$336$		0			0
$337$	−29.0000			−1.57973			−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000			−1.57973			−0.789865	+	0.613280i	$0.789850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−27.0000			−1.46213
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			5.19615i			0.278944i	0.990226	+	0.139472i	$0.0445405\pi$
$347$			5.19615i			0.278944i	−0.990226	+	0.139472i	$0.955459\pi$
$348$		0			0
$349$		−	27.7128i		−	1.48343i	−0.670714	−	0.741716i	$-0.734012\pi$
$349$		−	27.7128i		−	1.48343i	0.670714	−	0.741716i	$-0.265988\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−15.0000			−0.798369			−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000			−0.798369			−0.399185	+	0.916871i	$0.630707\pi$
$354$		0			0
$355$		−	15.5885i		−	0.827349i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			10.3923i			0.548485i	0.961661	+	0.274242i	$0.0884271\pi$
$359$			10.3923i			0.548485i	−0.961661	+	0.274242i	$0.911573\pi$
$360$		0			0
$361$	16.0000			0.842105
$362$		0			0
$363$		0			0
$364$		0			0
$365$			10.3923i			0.543958i
$366$		0			0
$367$		−	1.73205i		−	0.0904123i	−0.998978	−	0.0452062i	$-0.985606\pi$
$367$		−	1.73205i		−	0.0904123i	0.998978	−	0.0452062i	$-0.0143945\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	29.0000			1.50156			0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000			1.50156			0.750782	+	0.660551i	$0.229677\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	36.0000			1.85409
$378$		0			0
$379$	−10.0000			−0.513665			−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000			−0.513665			−0.256833	+	0.966456i	$0.582679\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000			0.613171			0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000			0.613171			0.306586	+	0.951843i	$0.400813\pi$
$384$		0			0
$385$	−27.0000	−	31.1769i	−1.37605	−	1.58892i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	20.7846i		−	1.05382i	−0.849921	−	0.526911i	$-0.823350\pi$
$389$		−	20.7846i		−	1.05382i	0.849921	−	0.526911i	$-0.176650\pi$
$390$		0			0
$391$			31.1769i			1.57668i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−42.0000			−2.11325
$396$		0			0
$397$			20.7846i			1.04315i	0.853206	+	0.521575i	$0.174655\pi$
$397$			20.7846i			1.04315i	−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	10.3923i		−	0.518967i	−0.965748	−	0.259483i	$-0.916448\pi$
$401$		−	10.3923i		−	0.518967i	0.965748	−	0.259483i	$-0.0835523\pi$
$402$		0			0
$403$	18.0000			0.896644
$404$		0			0
$405$		0			0
$406$		0			0
$407$			5.19615i			0.257564i
$408$		0			0
$409$		−	17.3205i		−	0.856444i	−0.903674	−	0.428222i	$-0.859140\pi$
$409$		−	17.3205i		−	0.856444i	0.903674	−	0.428222i	$-0.140860\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.0000	−	10.3923i	0.590481	−	0.511372i
$414$		0			0
$415$	−18.0000			−0.883585
$416$		0			0
$417$		0			0
$418$		0			0
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	5.00000			0.243685			0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000			0.243685			0.121843	+	0.992549i	$0.461120\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−24.0000			−1.16417
$426$		0			0
$427$	−24.0000	−	27.7128i	−1.16144	−	1.34112i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			36.3731i			1.75203i	0.482285	+	0.876014i	$0.339807\pi$
$431$			36.3731i			1.75203i	−0.482285	+	0.876014i	$0.660193\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	9.00000			0.430528
$438$		0			0
$439$			31.1769i			1.48799i	0.668184	+	0.743996i	$0.267072\pi$
$439$			31.1769i			1.48799i	−0.668184	+	0.743996i	$0.732928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	15.5885i		−	0.740630i	−0.928906	−	0.370315i	$-0.879250\pi$
$443$		−	15.5885i		−	0.740630i	0.928906	−	0.370315i	$-0.120750\pi$
$444$		0			0
$445$	27.0000			1.27992
$446$		0			0
$447$		0			0
$448$		0			0
$449$			20.7846i			0.980886i	0.871473	+	0.490443i	$0.163165\pi$
$449$			20.7846i			0.980886i	−0.871473	+	0.490443i	$0.836835\pi$
$450$		0			0
$451$		−	15.5885i		−	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	18.0000	+	20.7846i	0.843853	+	0.974398i
$456$		0			0
$457$	−17.0000			−0.795226			−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000			−0.795226			−0.397613	+	0.917553i	$0.630161\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.0000			0.698620			0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000			0.698620			0.349310	+	0.937007i	$0.386416\pi$
$462$		0			0
$463$	16.0000			0.743583			0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000			0.743583			0.371792	+	0.928316i	$0.378744\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.0000			0.832941			0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000			0.832941			0.416470	+	0.909149i	$0.363267\pi$
$468$		0			0
$469$	4.00000	−	3.46410i	0.184703	−	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	51.9615i		−	2.38919i
$474$		0			0
$475$			6.92820i			0.317888i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$		−	3.46410i		−	0.157949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			20.7846i			0.943781i
$486$		0			0
$487$	26.0000			1.17817			0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000			1.17817			0.589086	+	0.808070i	$0.299488\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	15.5885i		−	0.703497i	−0.936094	−	0.351749i	$-0.885587\pi$
$491$		−	15.5885i		−	0.703497i	0.936094	−	0.351749i	$-0.114413\pi$
$492$		0			0
$493$		−	62.3538i		−	2.80828i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.00000	+	10.3923i	0.403705	+	0.466159i
$498$		0			0
$499$	14.0000			0.626726			0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000			0.626726			0.313363	+	0.949633i	$0.398544\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6.00000			−0.267527			−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000			−0.267527			−0.133763	+	0.991013i	$0.542706\pi$
$504$		0			0
$505$	54.0000			2.40297
$506$		0			0
$507$		0			0
$508$		0			0
$509$	42.0000			1.86162			0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000			1.86162			0.930809	+	0.365507i	$0.119104\pi$
$510$		0			0
$511$	−6.00000	−	6.92820i	−0.265424	−	0.306486i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	25.9808i		−	1.14485i
$516$		0			0
$517$			31.1769i			1.37116i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	33.0000			1.44576			0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000			1.44576			0.722878	+	0.690976i	$0.242819\pi$
$522$		0			0
$523$		−	12.1244i		−	0.530161i	−0.964226	−	0.265081i	$-0.914601\pi$
$523$		−	12.1244i		−	0.530161i	0.964226	−	0.265081i	$-0.0853985\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	31.1769i		−	1.35809i
$528$		0			0
$529$	−4.00000			−0.173913
$530$		0			0
$531$		0			0
$532$		0			0
$533$			10.3923i			0.450141i
$534$		0			0
$535$		−	31.1769i		−	1.34790i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	36.0000	+	5.19615i	1.55063	+	0.223814i
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−33.0000			−1.41356
$546$		0			0
$547$	40.0000			1.71028			0.855138	−	0.518400i	$-0.173472\pi$
$547$	40.0000			1.71028			0.855138	+	0.518400i	$0.173472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.0000			−0.766826
$552$		0			0
$553$	28.0000	−	24.2487i	1.19068	−	1.03116i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	20.7846i		−	0.880672i	−0.897833	−	0.440336i	$-0.854859\pi$
$557$		−	20.7846i		−	0.880672i	0.897833	−	0.440336i	$-0.145141\pi$
$558$		0			0
$559$			34.6410i			1.46516i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.0000			0.758610			0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000			0.758610			0.379305	+	0.925272i	$0.376163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	41.5692i		−	1.74267i	−0.490687	−	0.871336i	$-0.663254\pi$
$569$		−	41.5692i		−	1.74267i	0.490687	−	0.871336i	$-0.336746\pi$
$570$		0			0
$571$	−20.0000			−0.836974			−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000			−0.836974			−0.418487	+	0.908223i	$0.637439\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	20.7846i		−	0.866778i
$576$		0			0
$577$			34.6410i			1.44212i	0.692870	+	0.721062i	$0.256346\pi$
$577$			34.6410i			1.44212i	−0.692870	+	0.721062i	$0.743654\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12.0000	−	10.3923i	0.497844	−	0.431145i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	36.0000			1.48588			0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000			1.48588			0.742940	+	0.669359i	$0.233431\pi$
$588$		0			0
$589$	−9.00000			−0.370839
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−39.0000			−1.60154			−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000			−1.60154			−0.800769	+	0.598973i	$0.795576\pi$
$594$		0			0
$595$	36.0000	−	31.1769i	1.47586	−	1.27813i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	25.9808i		−	1.06155i	−0.847514	−	0.530773i	$-0.821902\pi$
$599$		−	25.9808i		−	1.06155i	0.847514	−	0.530773i	$-0.178098\pi$
$600$		0			0
$601$			10.3923i			0.423911i	0.977279	+	0.211955i	$0.0679832\pi$
$601$			10.3923i			0.423911i	−0.977279	+	0.211955i	$0.932017\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−48.0000			−1.95148
$606$		0			0
$607$		−	17.3205i		−	0.703018i	−0.936185	−	0.351509i	$-0.885669\pi$
$607$		−	17.3205i		−	0.703018i	0.936185	−	0.351509i	$-0.114331\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	20.7846i		−	0.840855i
$612$		0			0
$613$	−19.0000			−0.767403			−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000			−0.767403			−0.383701	+	0.923457i	$0.625351\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			31.1769i			1.25514i	0.778562	+	0.627568i	$0.215949\pi$
$617$			31.1769i			1.25514i	−0.778562	+	0.627568i	$0.784051\pi$
$618$		0			0
$619$		−	32.9090i		−	1.32272i	−0.750067	−	0.661361i	$-0.769979\pi$
$619$		−	32.9090i		−	1.32272i	0.750067	−	0.661361i	$-0.230021\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.0000	+	15.5885i	−0.721155	+	0.624538i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.00000			−0.239236
$630$		0			0
$631$	−26.0000			−1.03504			−0.517522	−	0.855670i	$-0.673145\pi$
$631$	−26.0000			−1.03504			−0.517522	+	0.855670i	$0.673145\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.00000			−0.238103
$636$		0			0
$637$	−24.0000	−	3.46410i	−0.950915	−	0.137253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	20.7846i		−	0.820943i	−0.911873	−	0.410471i	$-0.865364\pi$
$641$		−	20.7846i		−	0.820943i	0.911873	−	0.410471i	$-0.134636\pi$
$642$		0			0
$643$		−	46.7654i		−	1.84425i	−0.386897	−	0.922123i	$-0.626453\pi$
$643$		−	46.7654i		−	1.84425i	0.386897	−	0.922123i	$-0.373547\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.0000			0.707653			0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000			0.707653			0.353827	+	0.935311i	$0.384880\pi$
$648$		0			0
$649$		−	31.1769i		−	1.22380i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			10.3923i			0.406682i	0.979108	+	0.203341i	$0.0651801\pi$
$653$			10.3923i			0.406682i	−0.979108	+	0.203341i	$0.934820\pi$
$654$		0			0
$655$	54.0000			2.10995
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	36.3731i		−	1.41689i	−0.705764	−	0.708447i	$-0.749396\pi$
$659$		−	36.3731i		−	1.41689i	0.705764	−	0.708447i	$-0.250604\pi$
$660$		0			0
$661$		−	24.2487i		−	0.943166i	−0.881822	−	0.471583i	$-0.843683\pi$
$661$		−	24.2487i		−	0.943166i	0.881822	−	0.471583i	$-0.156317\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−9.00000	−	10.3923i	−0.349005	−	0.402996i
$666$		0			0
$667$	54.0000			2.09089
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−72.0000			−2.77953
$672$		0			0
$673$	22.0000			0.848038			0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000			0.848038			0.424019	+	0.905653i	$0.360619\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	27.0000			1.03769			0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000			1.03769			0.518847	+	0.854867i	$0.326361\pi$
$678$		0			0
$679$	−12.0000	−	13.8564i	−0.460518	−	0.531760i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	15.5885i		−	0.596476i	−0.954492	−	0.298238i	$-0.903601\pi$
$683$		−	15.5885i		−	0.596476i	0.954492	−	0.298238i	$-0.0963989\pi$
$684$		0			0
$685$			31.1769i			1.19121i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	10.3923i		−	0.395342i	−0.980268	−	0.197671i	$-0.936662\pi$
$691$		−	10.3923i		−	0.395342i	0.980268	−	0.197671i	$-0.0633378\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			51.9615i			1.97101i
$696$		0			0
$697$	18.0000			0.681799
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	31.1769i		−	1.17754i	−0.808302	−	0.588768i	$-0.799613\pi$
$701$		−	31.1769i		−	1.17754i	0.808302	−	0.588768i	$-0.200387\pi$
$702$		0			0
$703$			1.73205i			0.0653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−36.0000	+	31.1769i	−1.35392	+	1.17253i
$708$		0			0
$709$	−23.0000			−0.863783			−0.431892	−	0.901926i	$-0.642154\pi$
$709$	−23.0000			−0.863783			−0.431892	+	0.901926i	$0.642154\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.0000			1.01116
$714$		0			0
$715$	54.0000			2.01949
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$		0			0
$721$	15.0000	+	17.3205i	0.558629	+	0.645049i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			41.5692i			1.54384i
$726$		0			0
$727$		−	31.1769i		−	1.15629i	−0.815935	−	0.578144i	$-0.803777\pi$
$727$		−	31.1769i		−	1.15629i	0.815935	−	0.578144i	$-0.196223\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	60.0000			2.21918
$732$		0			0
$733$		−	10.3923i		−	0.383849i	−0.981410	−	0.191924i	$-0.938527\pi$
$733$		−	10.3923i		−	0.383849i	0.981410	−	0.191924i	$-0.0614728\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	10.3923i		−	0.382805i
$738$		0			0
$739$	16.0000			0.588570			0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000			0.588570			0.294285	+	0.955718i	$0.404919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			36.3731i			1.33440i	0.744879	+	0.667199i	$0.232507\pi$
$743$			36.3731i			1.33440i	−0.744879	+	0.667199i	$0.767493\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000	+	20.7846i	0.657706	+	0.759453i
$750$		0			0
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−12.0000			−0.436725
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	22.0000	−	19.0526i	0.796453	−	0.689749i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			20.7846i			0.750489i
$768$		0			0
$769$			3.46410i			0.124919i	0.998048	+	0.0624593i	$0.0198944\pi$
$769$			3.46410i			0.124919i	−0.998048	+	0.0624593i	$0.980106\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	39.0000			1.40273			0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000			1.40273			0.701366	+	0.712801i	$0.252574\pi$
$774$		0			0
$775$			20.7846i			0.746605i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	5.19615i		−	0.186171i
$780$		0			0
$781$	27.0000			0.966136
$782$		0			0
$783$		0			0
$784$		0			0
$785$			31.1769i			1.11275i
$786$		0			0
$787$		−	38.1051i		−	1.35830i	−0.733999	−	0.679150i	$-0.762348\pi$
$787$		−	38.1051i		−	1.35830i	0.733999	−	0.679150i	$-0.237652\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	48.0000			1.70453
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−45.0000			−1.59398			−0.796991	−	0.603991i	$-0.793576\pi$
$797$	−45.0000			−1.59398			−0.796991	+	0.603991i	$0.793576\pi$
$798$		0			0
$799$	−36.0000			−1.27359
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.0000			−0.635206
$804$		0			0
$805$	27.0000	+	31.1769i	0.951625	+	1.09884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	31.1769i		−	1.09612i	−0.836438	−	0.548061i	$-0.815366\pi$
$809$		−	31.1769i		−	1.09612i	0.836438	−	0.548061i	$-0.184634\pi$
$810$		0			0
$811$		−	5.19615i		−	0.182462i	−0.995830	−	0.0912308i	$-0.970920\pi$
$811$		−	5.19615i		−	0.182462i	0.995830	−	0.0912308i	$-0.0290801\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	30.0000			1.05085
$816$		0			0
$817$		−	17.3205i		−	0.605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			31.1769i			1.08808i	0.839059	+	0.544041i	$0.183106\pi$
$821$			31.1769i			1.08808i	−0.839059	+	0.544041i	$0.816894\pi$
$822$		0			0
$823$	−20.0000			−0.697156			−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000			−0.697156			−0.348578	+	0.937280i	$0.613335\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			46.7654i			1.62619i	0.582130	+	0.813096i	$0.302219\pi$
$827$			46.7654i			1.62619i	−0.582130	+	0.813096i	$0.697781\pi$
$828$		0			0
$829$		−	45.0333i		−	1.56407i	−0.623233	−	0.782036i	$-0.714181\pi$
$829$		−	45.0333i		−	1.56407i	0.623233	−	0.782036i	$-0.285819\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.00000	+	41.5692i	−0.207888	+	1.44029i
$834$		0			0
$835$	−36.0000			−1.24583
$836$		0			0
$837$		0			0
$838$		0			0
$839$	42.0000			1.45000			0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000			1.45000			0.725001	+	0.688748i	$0.241839\pi$
$840$		0			0
$841$	−79.0000			−2.72414
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.00000			0.103203
$846$		0			0
$847$	32.0000	−	27.7128i	1.09953	−	0.952224i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	5.19615i		−	0.178122i
$852$		0			0
$853$			48.4974i			1.66052i	0.557376	+	0.830260i	$0.311808\pi$
$853$			48.4974i			1.66052i	−0.557376	+	0.830260i	$0.688192\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.0000			0.717346			0.358673	−	0.933463i	$-0.383229\pi$
$857$	21.0000			0.717346			0.358673	+	0.933463i	$0.383229\pi$
$858$		0			0
$859$			29.4449i			1.00465i	0.864680	+	0.502323i	$0.167521\pi$
$859$			29.4449i			1.00465i	−0.864680	+	0.502323i	$0.832479\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			10.3923i			0.353758i	0.984233	+	0.176879i	$0.0566002\pi$
$863$			10.3923i			0.353758i	−0.984233	+	0.176879i	$0.943400\pi$
$864$		0			0
$865$	−27.0000			−0.918028
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	72.7461i		−	2.46774i
$870$		0			0
$871$			6.92820i			0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.00000	−	5.19615i	0.202837	−	0.175662i
$876$		0			0
$877$	2.00000			0.0675352			0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000			0.0675352			0.0337676	+	0.999430i	$0.489249\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.0000			−0.909653			−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000			−0.909653			−0.454827	+	0.890580i	$0.650299\pi$
$882$		0			0
$883$	−28.0000			−0.942275			−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000			−0.942275			−0.471138	+	0.882060i	$0.656156\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	42.0000			1.41022			0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000			1.41022			0.705111	+	0.709097i	$0.250897\pi$
$888$		0			0
$889$	4.00000	−	3.46410i	0.134156	−	0.116182i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			10.3923i			0.347765i
$894$		0			0
$895$			31.1769i			1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−54.0000			−1.80100
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			20.7846i			0.690904i
$906$		0			0
$907$	32.0000			1.06254			0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000			1.06254			0.531271	+	0.847202i	$0.321714\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	10.3923i		−	0.344312i	−0.985070	−	0.172156i	$-0.944927\pi$
$911$		−	10.3923i		−	0.344312i	0.985070	−	0.172156i	$-0.0550734\pi$
$912$		0			0
$913$		−	31.1769i		−	1.03181i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−36.0000	+	31.1769i	−1.18882	+	1.02955i
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−18.0000			−0.592477
$924$		0			0
$925$	4.00000			0.131519
$926$		0			0
$927$		0			0
$928$		0			0
$929$	30.0000			0.984268			0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000			0.984268			0.492134	+	0.870519i	$0.336217\pi$
$930$		0			0
$931$	12.0000	+	1.73205i	0.393284	+	0.0567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	93.5307i		−	3.05878i
$936$		0			0
$937$			55.4256i			1.81068i	0.424691	+	0.905338i	$0.360383\pi$
$937$			55.4256i			1.81068i	−0.424691	+	0.905338i	$0.639617\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.0000			−0.684580			−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000			−0.684580			−0.342290	+	0.939594i	$0.611203\pi$
$942$		0			0
$943$			15.5885i			0.507630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			25.9808i			0.844261i	0.906535	+	0.422131i	$0.138718\pi$
$947$			25.9808i			0.844261i	−0.906535	+	0.422131i	$0.861282\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$		0			0
$951$		0			0
$952$		0			0
$953$			31.1769i			1.00992i	0.863143	+	0.504960i	$0.168493\pi$
$953$			31.1769i			1.00992i	−0.863143	+	0.504960i	$0.831507\pi$
$954$		0			0
$955$		−	15.5885i		−	0.504431i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000	−	20.7846i	−0.581250	−	0.671170i
$960$		0			0
$961$	4.00000			0.129032
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.00000			−0.193147
$966$		0			0
$967$	16.0000			0.514525			0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000			0.514525			0.257263	+	0.966342i	$0.417179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.00000			−0.192549			−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000			−0.192549			−0.0962746	+	0.995355i	$0.530693\pi$
$972$		0			0
$973$	−30.0000	−	34.6410i	−0.961756	−	1.11054i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	41.5692i		−	1.32992i	−0.746880	−	0.664959i	$-0.768449\pi$
$977$		−	41.5692i		−	1.32992i	0.746880	−	0.664959i	$-0.231551\pi$
$978$		0			0
$979$			46.7654i			1.49463i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−42.0000			−1.33959			−0.669796	−	0.742545i	$-0.733618\pi$
$983$	−42.0000			−1.33959			−0.669796	+	0.742545i	$0.733618\pi$
$984$		0			0
$985$		−	62.3538i		−	1.98676i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			51.9615i			1.65228i
$990$		0			0
$991$	−10.0000			−0.317660			−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000			−0.317660			−0.158830	+	0.987306i	$0.550772\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5.19615i		−	0.164729i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.k.d.1889.2		2
3.2	odd	2		3024.2.k.a.1889.2		2
4.3	odd	2		756.2.f.c.377.1	yes	2
7.6	odd	2		3024.2.k.a.1889.1		2
12.11	even	2		756.2.f.a.377.1	✓	2
21.20	even	2	inner	3024.2.k.d.1889.1		2
28.27	even	2		756.2.f.a.377.2	yes	2
36.7	odd	6		2268.2.x.a.377.1		2
36.11	even	6		2268.2.x.g.377.1		2
36.23	even	6		2268.2.x.h.1889.1		2
36.31	odd	6		2268.2.x.b.1889.1		2
84.83	odd	2		756.2.f.c.377.2	yes	2
252.83	odd	6		2268.2.x.b.377.1		2
252.139	even	6		2268.2.x.g.1889.1		2
252.167	odd	6		2268.2.x.a.1889.1		2
252.223	even	6		2268.2.x.h.377.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.f.a.377.1	✓	2	12.11	even	2
756.2.f.a.377.2	yes	2	28.27	even	2
756.2.f.c.377.1	yes	2	4.3	odd	2
756.2.f.c.377.2	yes	2	84.83	odd	2
2268.2.x.a.377.1		2	36.7	odd	6
2268.2.x.a.1889.1		2	252.167	odd	6
2268.2.x.b.377.1		2	252.83	odd	6
2268.2.x.b.1889.1		2	36.31	odd	6
2268.2.x.g.377.1		2	36.11	even	6
2268.2.x.g.1889.1		2	252.139	even	6
2268.2.x.h.377.1		2	252.223	even	6
2268.2.x.h.1889.1		2	36.23	even	6
3024.2.k.a.1889.1		2	7.6	odd	2
3024.2.k.a.1889.2		2	3.2	odd	2
3024.2.k.d.1889.1		2	21.20	even	2	inner
3024.2.k.d.1889.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{5} +(-2.00000 + 1.73205i) q^{7} +5.19615i q^{11} -3.46410i q^{13} -6.00000 q^{17} +1.73205i q^{19} -5.19615i q^{23} +4.00000 q^{25} +10.3923i q^{29} +5.19615i q^{31} +(-6.00000 + 5.19615i) q^{35} +1.00000 q^{37} -3.00000 q^{41} -10.0000 q^{43} +6.00000 q^{47} +(1.00000 - 6.92820i) q^{49} +15.5885i q^{55} -6.00000 q^{59} +13.8564i q^{61} -10.3923i q^{65} -2.00000 q^{67} -5.19615i q^{71} +3.46410i q^{73} +(-9.00000 - 10.3923i) q^{77} -14.0000 q^{79} -6.00000 q^{83} -18.0000 q^{85} +9.00000 q^{89} +(6.00000 + 6.92820i) q^{91} +5.19615i q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} - 4 q^{7} - 12 q^{17} + 8 q^{25} - 12 q^{35} + 2 q^{37} - 6 q^{41} - 20 q^{43} + 12 q^{47} + 2 q^{49} - 12 q^{59} - 4 q^{67} - 18 q^{77} - 28 q^{79} - 12 q^{83} - 36 q^{85} + 18 q^{89} + 12 q^{91}+O(q^{100}) \) 2 * q + 6 * q^5 - 4 * q^7 - 12 * q^17 + 8 * q^25 - 12 * q^35 + 2 * q^37 - 6 * q^41 - 20 * q^43 + 12 * q^47 + 2 * q^49 - 12 * q^59 - 4 * q^67 - 18 * q^77 - 28 * q^79 - 12 * q^83 - 36 * q^85 + 18 * q^89 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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