LMFDB - Embedded newform 2240.2.e.c.2239.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(2239,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.e (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:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2239.4
Root			$1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2240.2239
Dual form			2240.2.e.c.2239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{3} +2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.41421i q^{3} +2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} +1.00000 q^{9} -3.16228 q^{15} +(3.00000 - 2.23607i) q^{21} -9.48683 q^{23} -5.00000 q^{25} +5.65685i q^{27} -6.00000 q^{29} +(4.74342 - 3.53553i) q^{35} +4.47214i q^{41} -3.16228 q^{43} +2.23607i q^{45} -9.89949i q^{47} +(-2.00000 + 6.70820i) q^{49} +13.4164i q^{61} +(-1.58114 - 2.12132i) q^{63} -15.8114 q^{67} -13.4164i q^{69} -7.07107i q^{75} -5.00000 q^{81} -15.5563i q^{83} -8.48528i q^{87} -17.8885i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} + 12 q^{21} - 20 q^{25} - 24 q^{29} - 8 q^{49} - 20 q^{81}+O(q^{100}) $ 4 * q + 4 * q^9 + 12 * q^21 - 20 * q^25 - 24 * q^29 - 8 * q^49 - 20 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$.

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$			1.41421i			0.816497i	0.912871	+	0.408248i	$0.133860\pi$
$3$			1.41421i			0.816497i	−0.912871	+	0.408248i	$0.866140\pi$
$4$		0			0
$5$			2.23607i			1.00000i
$5$			2.23607i			1.00000i
$6$		0			0
$7$	−1.58114	−	2.12132i	−0.597614	−	0.801784i
$7$	−1.58114	−	2.12132i	−0.597614	−	0.801784i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−3.16228			−0.816497
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	3.00000	−	2.23607i	0.654654	−	0.487950i
$22$		0			0
$23$	−9.48683			−1.97814			−0.989071	−	0.147442i	$-0.952896\pi$
$23$	−9.48683			−1.97814			−0.989071	+	0.147442i	$0.952896\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$			5.65685i			1.08866i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.74342	−	3.53553i	0.801784	−	0.597614i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$	−3.16228			−0.482243			−0.241121	−	0.970495i	$-0.577515\pi$
$43$	−3.16228			−0.482243			−0.241121	+	0.970495i	$0.577515\pi$
$44$		0			0
$45$			2.23607i			0.333333i
$46$		0			0
$47$		−	9.89949i		−	1.44399i	−0.691898	−	0.721995i	$-0.743225\pi$
$47$		−	9.89949i		−	1.44399i	0.691898	−	0.721995i	$-0.256775\pi$
$48$		0			0
$49$	−2.00000	+	6.70820i	−0.285714	+	0.958315i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	−1.58114	−	2.12132i	−0.199205	−	0.267261i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−15.8114			−1.93167			−0.965834	−	0.259161i	$-0.916554\pi$
$67$	−15.8114			−1.93167			−0.965834	+	0.259161i	$0.916554\pi$
$68$		0			0
$69$		−	13.4164i		−	1.61515i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		−	7.07107i		−	0.816497i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$		−	15.5563i		−	1.70753i	−0.520658	−	0.853766i	$-0.674313\pi$
$83$		−	15.5563i		−	1.70753i	0.520658	−	0.853766i	$-0.325687\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	8.48528i		−	0.909718i
$88$		0			0
$89$		−	17.8885i		−	1.89618i	−0.317999	−	0.948091i	$-0.603011\pi$
$89$		−	17.8885i		−	1.89618i	0.317999	−	0.948091i	$-0.396989\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.94427i			0.889988i	0.895533	+	0.444994i	$0.146794\pi$
$101$			8.94427i			0.889988i	−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$			12.7279i			1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$			12.7279i			1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$		0			0
$105$	5.00000	+	6.70820i	0.487950	+	0.654654i
$106$		0			0
$107$	−9.48683			−0.917127			−0.458563	−	0.888662i	$-0.651636\pi$
$107$	−9.48683			−0.917127			−0.458563	+	0.888662i	$0.651636\pi$
$108$		0			0
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	21.2132i		−	1.97814i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	−6.32456			−0.570266
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$	22.1359			1.96425			0.982124	−	0.188237i	$-0.0602772\pi$
$127$	22.1359			1.96425			0.982124	+	0.188237i	$0.0602772\pi$
$128$		0			0
$129$		−	4.47214i		−	0.393750i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−12.6491			−1.08866
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	14.0000			1.17901
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	13.4164i		−	1.11417i
$146$		0			0
$147$	−9.48683	−	2.82843i	−0.782461	−	0.233285i
$148$		0			0
$149$	24.0000			1.96616			0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000			1.96616			0.983078	+	0.183186i	$0.0586410\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	15.0000	+	20.1246i	1.18217	+	1.58604i
$162$		0			0
$163$	−22.1359			−1.73382			−0.866910	−	0.498464i	$-0.833898\pi$
$163$	−22.1359			−1.73382			−0.866910	+	0.498464i	$0.833898\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	24.0416i		−	1.86040i	−0.367057	−	0.930199i	$-0.619634\pi$
$167$		−	24.0416i		−	1.86040i	0.367057	−	0.930199i	$-0.380366\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	7.90569	+	10.6066i	0.597614	+	0.801784i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	26.8328i		−	1.99447i	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$		−	26.8328i		−	1.99447i	0.0743294	−	0.997234i	$-0.476318\pi$
$182$		0			0
$183$	−18.9737			−1.40257
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	12.0000	−	8.94427i	0.872872	−	0.650600i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	22.3607i		−	1.57720i
$202$		0			0
$203$	9.48683	+	12.7279i	0.665845	+	0.893325i
$204$		0			0
$205$	−10.0000			−0.698430
$206$		0			0
$207$	−9.48683			−0.659380
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	7.07107i		−	0.482243i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			29.6985i			1.98876i	0.105881	+	0.994379i	$0.466234\pi$
$223$			29.6985i			1.98876i	−0.105881	+	0.994379i	$0.533766\pi$
$224$		0			0
$225$	−5.00000			−0.333333
$226$		0			0
$227$			9.89949i			0.657053i	0.944495	+	0.328526i	$0.106552\pi$
$227$			9.89949i			0.657053i	−0.944495	+	0.328526i	$0.893448\pi$
$228$		0			0
$229$			26.8328i			1.77316i	0.462573	+	0.886581i	$0.346926\pi$
$229$			26.8328i			1.77316i	−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	22.1359			1.44399
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			13.4164i			0.864227i	0.901819	+	0.432113i	$0.142232\pi$
$241$			13.4164i			0.864227i	−0.901819	+	0.432113i	$0.857768\pi$
$242$		0			0
$243$			9.89949i			0.635053i
$244$		0			0
$245$	−15.0000	−	4.47214i	−0.958315	−	0.285714i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	22.0000			1.39419
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	−28.4605			−1.75495			−0.877475	−	0.479623i	$-0.840774\pi$
$263$	−28.4605			−1.75495			−0.877475	+	0.479623i	$0.840774\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	25.2982			1.54823
$268$		0			0
$269$			22.3607i			1.36335i	0.731653	+	0.681677i	$0.238749\pi$
$269$			22.3607i			1.36335i	−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$			29.6985i			1.76539i	0.469945	+	0.882696i	$0.344274\pi$
$283$			29.6985i			1.76539i	−0.469945	+	0.882696i	$0.655726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.48683	−	7.07107i	0.559990	−	0.417392i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	5.00000	+	6.70820i	0.288195	+	0.386654i
$302$		0			0
$303$	−12.6491			−0.726672
$304$		0			0
$305$	−30.0000			−1.71780
$306$		0			0
$307$			4.24264i			0.242140i	0.992644	+	0.121070i	$0.0386326\pi$
$307$			4.24264i			0.242140i	−0.992644	+	0.121070i	$0.961367\pi$
$308$		0			0
$309$	−18.0000			−1.02398
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$	4.74342	−	3.53553i	0.267261	−	0.199205i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		−	13.4164i		−	0.748831i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$		−	22.6274i		−	1.25130i
$328$		0			0
$329$	−21.0000	+	15.6525i	−1.15777	+	0.862949i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	35.3553i		−	1.93167i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	17.3925	−	6.36396i	0.939108	−	0.343622i
$344$		0			0
$345$	30.0000			1.61515
$346$		0			0
$347$	28.4605			1.52784			0.763920	−	0.645311i	$-0.223272\pi$
$347$	28.4605			1.52784			0.763920	+	0.645311i	$0.223272\pi$
$348$		0			0
$349$		−	26.8328i		−	1.43633i	−0.695874	−	0.718164i	$-0.744983\pi$
$349$		−	26.8328i		−	1.43633i	0.695874	−	0.718164i	$-0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$			15.5563i			0.816497i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			38.1838i			1.99318i	0.0825348	+	0.996588i	$0.473698\pi$
$367$			38.1838i			1.99318i	−0.0825348	+	0.996588i	$0.526302\pi$
$368$		0			0
$369$			4.47214i			0.232810i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	15.8114			0.816497
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$			31.3050i			1.60380i
$382$		0			0
$383$			26.8701i			1.37300i	0.727132	+	0.686498i	$0.240853\pi$
$383$			26.8701i			1.37300i	−0.727132	+	0.686498i	$0.759147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.16228			−0.160748
$388$		0			0
$389$	−24.0000			−1.21685			−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000			−1.21685			−0.608424	+	0.793612i	$0.708198\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	11.1803i		−	0.555556i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			40.2492i			1.99020i	0.0988936	+	0.995098i	$0.468470\pi$
$409$			40.2492i			1.99020i	−0.0988936	+	0.995098i	$0.531530\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	34.7851			1.70753
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$		−	9.89949i		−	0.481330i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	28.4605	−	21.2132i	1.37730	−	1.02658i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$	18.9737			0.909718
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−2.00000	+	6.70820i	−0.0952381	+	0.319438i
$442$		0			0
$443$	9.48683			0.450733			0.225367	−	0.974274i	$-0.427642\pi$
$443$	9.48683			0.450733			0.225367	+	0.974274i	$0.427642\pi$
$444$		0			0
$445$	40.0000			1.89618
$446$		0			0
$447$			33.9411i			1.60536i
$448$		0			0
$449$	36.0000			1.69895			0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000			1.69895			0.849473	+	0.527633i	$0.176920\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	8.94427i		−	0.416576i	−0.978068	−	0.208288i	$-0.933211\pi$
$461$		−	8.94427i		−	0.416576i	0.978068	−	0.208288i	$-0.0667892\pi$
$462$		0			0
$463$	41.1096			1.91053			0.955263	−	0.295758i	$-0.0955723\pi$
$463$	41.1096			1.91053			0.955263	+	0.295758i	$0.0955723\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	32.5269i		−	1.50517i	−0.658497	−	0.752583i	$-0.728808\pi$
$467$		−	32.5269i		−	1.50517i	0.658497	−	0.752583i	$-0.271192\pi$
$468$		0			0
$469$	25.0000	+	33.5410i	1.15439	+	1.54878i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−28.4605	+	21.2132i	−1.29500	+	0.965234i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.1359			1.00308			0.501538	−	0.865136i	$-0.332768\pi$
$487$	22.1359			1.00308			0.501538	+	0.865136i	$0.332768\pi$
$488$		0			0
$489$		−	31.3050i		−	1.41566i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	34.0000			1.51901
$502$		0			0
$503$			43.8406i			1.95476i	0.211498	+	0.977378i	$0.432166\pi$
$503$			43.8406i			1.95476i	−0.211498	+	0.977378i	$0.567834\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$		−	18.3848i		−	0.816497i
$508$		0			0
$509$		−	44.7214i		−	1.98224i	−0.132973	−	0.991120i	$-0.542452\pi$
$509$		−	44.7214i		−	1.98224i	0.132973	−	0.991120i	$-0.457548\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28.4605			−1.25412
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.8885i			0.783711i	0.920027	+	0.391856i	$0.128167\pi$
$521$			17.8885i			0.783711i	−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$			29.6985i			1.29862i	0.760522	+	0.649312i	$0.224943\pi$
$523$			29.6985i			1.29862i	−0.760522	+	0.649312i	$0.775057\pi$
$524$		0			0
$525$	−15.0000	+	11.1803i	−0.654654	+	0.487950i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	67.0000			2.91304
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	21.2132i		−	0.917127i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−38.0000			−1.63375			−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000			−1.63375			−0.816874	+	0.576816i	$0.804295\pi$
$542$		0			0
$543$	37.9473			1.62848
$544$		0			0
$545$		−	35.7771i		−	1.53252i
$546$		0			0
$547$	3.16228			0.135209			0.0676046	−	0.997712i	$-0.478464\pi$
$547$	3.16228			0.135209			0.0676046	+	0.997712i	$0.478464\pi$
$548$		0			0
$549$			13.4164i			0.572598i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.41421i		−	0.0596020i	−0.999556	−	0.0298010i	$-0.990513\pi$
$563$		−	1.41421i		−	0.0596020i	0.999556	−	0.0298010i	$-0.00948736\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.90569	+	10.6066i	0.332008	+	0.445435i
$568$		0			0
$569$	−36.0000			−1.50920			−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000			−1.50920			−0.754599	+	0.656186i	$0.772169\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	47.4342			1.97814
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−33.0000	+	24.5967i	−1.36907	+	1.02045i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	9.89949i		−	0.408596i	−0.978909	−	0.204298i	$-0.934509\pi$
$587$		−	9.89949i		−	0.408596i	0.978909	−	0.204298i	$-0.0654911\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	40.2492i		−	1.64180i	−0.571072	−	0.820900i	$-0.693472\pi$
$601$		−	40.2492i		−	1.64180i	0.571072	−	0.820900i	$-0.306528\pi$
$602$		0			0
$603$	−15.8114			−0.643890
$604$		0			0
$605$			24.5967i			1.00000i
$606$		0			0
$607$			46.6690i			1.89424i	0.320882	+	0.947119i	$0.396021\pi$
$607$			46.6690i			1.89424i	−0.320882	+	0.947119i	$0.603979\pi$
$608$		0			0
$609$	−18.0000	+	13.4164i	−0.729397	+	0.543660i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		−	14.1421i		−	0.570266i
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		−	53.6656i		−	2.15353i
$622$		0			0
$623$	−37.9473	+	28.2843i	−1.52033	+	1.13319i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			49.4975i			1.96425i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000			−0.473972			−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000			−0.473972			−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$			29.6985i			1.17119i	0.810602	+	0.585597i	$0.199140\pi$
$643$			29.6985i			1.17119i	−0.810602	+	0.585597i	$0.800860\pi$
$644$		0			0
$645$	10.0000			0.393750
$646$		0			0
$647$			18.3848i			0.722780i	0.932415	+	0.361390i	$0.117698\pi$
$647$			18.3848i			0.722780i	−0.932415	+	0.361390i	$0.882302\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	40.2492i		−	1.56551i	−0.622328	−	0.782757i	$-0.713813\pi$
$661$		−	40.2492i		−	1.56551i	0.622328	−	0.782757i	$-0.286187\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	56.9210			2.20399
$668$		0			0
$669$	−42.0000			−1.62381
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	28.2843i		−	1.08866i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−14.0000			−0.536481
$682$		0			0
$683$	−28.4605			−1.08901			−0.544505	−	0.838757i	$-0.683283\pi$
$683$	−28.4605			−1.08901			−0.544505	+	0.838757i	$0.683283\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−37.9473			−1.44778
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$			31.3050i			1.17901i
$706$		0			0
$707$	18.9737	−	14.1421i	0.713578	−	0.531870i
$708$		0			0
$709$	46.0000			1.72757			0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000			1.72757			0.863783	+	0.503864i	$0.168089\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	27.0000	−	20.1246i	1.00553	−	0.749480i
$722$		0			0
$723$	−18.9737			−0.705638
$724$		0			0
$725$	30.0000			1.11417
$726$		0			0
$727$		−	4.24264i		−	0.157351i	−0.996900	−	0.0786754i	$-0.974931\pi$
$727$		−	4.24264i		−	0.157351i	0.996900	−	0.0786754i	$-0.0250691\pi$
$728$		0			0
$729$	−29.0000			−1.07407
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	6.32456	−	21.2132i	0.233285	−	0.782461i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−47.4342			−1.74019			−0.870095	−	0.492883i	$-0.835943\pi$
$743$	−47.4342			−1.74019			−0.870095	+	0.492883i	$0.835943\pi$
$744$		0			0
$745$			53.6656i			1.96616i
$746$		0			0
$747$		−	15.5563i		−	0.569177i
$748$		0			0
$749$	15.0000	+	20.1246i	0.548088	+	0.735337i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			35.7771i			1.29692i	0.761249	+	0.648459i	$0.224586\pi$
$761$			35.7771i			1.29692i	−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$	25.2982	+	33.9411i	0.915857	+	1.22875i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	53.6656i		−	1.93523i	−0.252426	−	0.967616i	$-0.581229\pi$
$769$		−	53.6656i		−	1.93523i	0.252426	−	0.967616i	$-0.418771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	33.9411i		−	1.21296i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			38.1838i			1.36110i	0.732700	+	0.680552i	$0.238260\pi$
$787$			38.1838i			1.36110i	−0.732700	+	0.680552i	$0.761740\pi$
$788$		0			0
$789$		−	40.2492i		−	1.43291i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	17.8885i		−	0.632061i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−45.0000	+	33.5410i	−1.58604	+	1.18217i
$806$		0			0
$807$	−31.6228			−1.11317
$808$		0			0
$809$	54.0000			1.89854			0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000			1.89854			0.949269	+	0.314464i	$0.101825\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	49.4975i		−	1.73382i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−48.0000			−1.67521			−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000			−1.67521			−0.837606	+	0.546275i	$0.816045\pi$
$822$		0			0
$823$	−15.8114			−0.551150			−0.275575	−	0.961280i	$-0.588868\pi$
$823$	−15.8114			−0.551150			−0.275575	+	0.961280i	$0.588868\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−47.4342			−1.64945			−0.824724	−	0.565536i	$-0.808669\pi$
$827$	−47.4342			−1.64945			−0.824724	+	0.565536i	$0.808669\pi$
$828$		0			0
$829$		−	13.4164i		−	0.465971i	−0.972480	−	0.232986i	$-0.925151\pi$
$829$		−	13.4164i		−	0.465971i	0.972480	−	0.232986i	$-0.0748495\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	53.7587			1.86040
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$			16.9706i			0.584497i
$844$		0			0
$845$		−	29.0689i		−	1.00000i
$846$		0			0
$847$	−17.3925	−	23.3345i	−0.597614	−	0.801784i
$848$		0			0
$849$	−42.0000			−1.44144
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$	10.0000	+	13.4164i	0.340799	+	0.457230i
$862$		0			0
$863$	9.48683			0.322936			0.161468	−	0.986878i	$-0.448377\pi$
$863$	9.48683			0.322936			0.161468	+	0.986878i	$0.448377\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	24.0416i		−	0.816497i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−23.7171	+	17.6777i	−0.801784	+	0.597614i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	58.1378i		−	1.95871i	−0.202145	−	0.979356i	$-0.564791\pi$
$881$		−	58.1378i		−	1.95871i	0.202145	−	0.979356i	$-0.435209\pi$
$882$		0			0
$883$	22.1359			0.744934			0.372467	−	0.928045i	$-0.378512\pi$
$883$	22.1359			0.744934			0.372467	+	0.928045i	$0.378512\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	52.3259i		−	1.75693i	−0.477805	−	0.878466i	$-0.658567\pi$
$887$		−	52.3259i		−	1.75693i	0.477805	−	0.878466i	$-0.341433\pi$
$888$		0			0
$889$	−35.0000	−	46.9574i	−1.17386	−	1.57490i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−9.48683	+	7.07107i	−0.315702	+	0.235310i
$904$		0			0
$905$	60.0000			1.99447
$906$		0			0
$907$	60.0833			1.99503			0.997516	−	0.0704373i	$-0.0224395\pi$
$907$	60.0833			1.99503			0.997516	+	0.0704373i	$0.0224395\pi$
$908$		0			0
$909$			8.94427i			0.296663i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	42.4264i		−	1.40257i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−6.00000			−0.197707
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$			12.7279i			0.418040i
$928$		0			0
$929$			49.1935i			1.61399i	0.590561	+	0.806993i	$0.298907\pi$
$929$			49.1935i			1.61399i	−0.590561	+	0.806993i	$0.701093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			44.7214i			1.45787i	0.684580	+	0.728937i	$0.259985\pi$
$941$			44.7214i			1.45787i	−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$		−	42.4264i		−	1.38159i
$944$		0			0
$945$	20.0000	+	26.8328i	0.650600	+	0.872872i
$946$		0			0
$947$	−9.48683			−0.308281			−0.154140	−	0.988049i	$-0.549261\pi$
$947$	−9.48683			−0.308281			−0.154140	+	0.988049i	$0.549261\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	−9.48683			−0.305709
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−41.1096			−1.32200			−0.660998	−	0.750388i	$-0.729867\pi$
$967$	−41.1096			−1.32200			−0.660998	+	0.750388i	$0.729867\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−16.0000			−0.510841
$982$		0			0
$983$			41.0122i			1.30809i	0.756457	+	0.654043i	$0.226928\pi$
$983$			41.0122i			1.30809i	−0.756457	+	0.654043i	$0.773072\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−22.1359	−	29.6985i	−0.704595	−	0.945313i
$988$		0			0
$989$	30.0000			0.953945
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.e.c.2239.4		4
4.3	odd	2	inner	2240.2.e.c.2239.2		4
5.4	even	2	inner	2240.2.e.c.2239.2		4
7.6	odd	2	inner	2240.2.e.c.2239.1		4
8.3	odd	2		560.2.e.b.559.3	yes	4
8.5	even	2		560.2.e.b.559.1	✓	4
20.19	odd	2	CM	2240.2.e.c.2239.4		4
28.27	even	2	inner	2240.2.e.c.2239.3		4
35.34	odd	2	inner	2240.2.e.c.2239.3		4
40.3	even	4		2800.2.k.g.2351.1		4
40.13	odd	4		2800.2.k.g.2351.4		4
40.19	odd	2		560.2.e.b.559.1	✓	4
40.27	even	4		2800.2.k.g.2351.4		4
40.29	even	2		560.2.e.b.559.3	yes	4
40.37	odd	4		2800.2.k.g.2351.1		4
56.13	odd	2		560.2.e.b.559.4	yes	4
56.27	even	2		560.2.e.b.559.2	yes	4
140.139	even	2	inner	2240.2.e.c.2239.1		4
280.13	even	4		2800.2.k.g.2351.2		4
280.27	odd	4		2800.2.k.g.2351.2		4
280.69	odd	2		560.2.e.b.559.2	yes	4
280.83	odd	4		2800.2.k.g.2351.3		4
280.139	even	2		560.2.e.b.559.4	yes	4
280.237	even	4		2800.2.k.g.2351.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.e.b.559.1	✓	4	8.5	even	2
560.2.e.b.559.1	✓	4	40.19	odd	2
560.2.e.b.559.2	yes	4	56.27	even	2
560.2.e.b.559.2	yes	4	280.69	odd	2
560.2.e.b.559.3	yes	4	8.3	odd	2
560.2.e.b.559.3	yes	4	40.29	even	2
560.2.e.b.559.4	yes	4	56.13	odd	2
560.2.e.b.559.4	yes	4	280.139	even	2
2240.2.e.c.2239.1		4	7.6	odd	2	inner
2240.2.e.c.2239.1		4	140.139	even	2	inner
2240.2.e.c.2239.2		4	4.3	odd	2	inner
2240.2.e.c.2239.2		4	5.4	even	2	inner
2240.2.e.c.2239.3		4	28.27	even	2	inner
2240.2.e.c.2239.3		4	35.34	odd	2	inner
2240.2.e.c.2239.4		4	1.1	even	1	trivial
2240.2.e.c.2239.4		4	20.19	odd	2	CM
2800.2.k.g.2351.1		4	40.3	even	4
2800.2.k.g.2351.1		4	40.37	odd	4
2800.2.k.g.2351.2		4	280.13	even	4
2800.2.k.g.2351.2		4	280.27	odd	4
2800.2.k.g.2351.3		4	280.83	odd	4
2800.2.k.g.2351.3		4	280.237	even	4
2800.2.k.g.2351.4		4	40.13	odd	4
2800.2.k.g.2351.4		4	40.27	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{3} +2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{3} +2.23607i q^{5} +(-1.58114 - 2.12132i) q^{7} +1.00000 q^{9} -3.16228 q^{15} +(3.00000 - 2.23607i) q^{21} -9.48683 q^{23} -5.00000 q^{25} +5.65685i q^{27} -6.00000 q^{29} +(4.74342 - 3.53553i) q^{35} +4.47214i q^{41} -3.16228 q^{43} +2.23607i q^{45} -9.89949i q^{47} +(-2.00000 + 6.70820i) q^{49} +13.4164i q^{61} +(-1.58114 - 2.12132i) q^{63} -15.8114 q^{67} -13.4164i q^{69} -7.07107i q^{75} -5.00000 q^{81} -15.5563i q^{83} -8.48528i q^{87} -17.8885i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} + 12 q^{21} - 20 q^{25} - 24 q^{29} - 8 q^{49} - 20 q^{81}+O(q^{100}) \) 4 * q + 4 * q^9 + 12 * q^21 - 20 * q^25 - 24 * q^29 - 8 * q^49 - 20 * q^81

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