LMFDB - Embedded newform 2240.2.n.i.1119.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1119,2240)]

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, 1, 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("2240.1119");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.n (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.101415451701035401216.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 $ x^16 - 18x^12 + 145x^8 - 72*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1119.14
Root			$0.752908 - 0.137538i$ of defining polynomial
Character	$\chi$	$=$	2240.1119
Dual form			2240.2.n.i.1119.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.97127 q^{3} +(-2.14973 + 0.615370i) q^{5} +2.64575 q^{7} +5.82843 q^{9} +O(q^{10})$	$q+2.97127 q^{3} +(-2.14973 + 0.615370i) q^{5} +2.64575 q^{7} +5.82843 q^{9} +7.17327i q^{13} +(-6.38741 + 1.82843i) q^{15} +6.81801i q^{19} +7.86123 q^{21} -7.48331 q^{23} +(4.24264 - 2.64575i) q^{25} +8.40401 q^{27} +(-5.68764 + 1.62811i) q^{35} +21.3137i q^{39} +(-12.5295 + 3.58664i) q^{45} +7.00000 q^{49} +20.2581i q^{57} -13.9416i q^{59} +11.4230 q^{61} +15.4206 q^{63} +(-4.41421 - 15.4206i) q^{65} -22.2349 q^{69} +5.65685i q^{71} +(12.6060 - 7.86123i) q^{75} -16.9706i q^{79} +7.48528 q^{81} +13.8368 q^{83} +18.9787i q^{91} +(-4.19560 - 14.6569i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9}+O(q^{10}) $ 16 * q + 48 * q^9	$ 16 q + 48 q^{9} + 112 q^{49} - 48 q^{65} - 16 q^{81}+O(q^{100}) $ 16 * q + 48 * q^9 + 112 * q^49 - 48 * q^65 - 16 * 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
$3$	2.97127			1.71546			0.857731	−	0.514099i	$-0.171874\pi$
$3$	2.97127			1.71546			0.857731	+	0.514099i	$0.171874\pi$
$4$		0			0
$5$	−2.14973	+	0.615370i	−0.961387	+	0.275202i
$5$	−2.14973	+	0.615370i	−0.961387	+	0.275202i
$6$		0			0
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$		0			0
$9$	5.82843			1.94281
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			7.17327i			1.98951i	0.102296	+	0.994754i	$0.467381\pi$
$13$			7.17327i			1.98951i	−0.102296	+	0.994754i	$0.532619\pi$
$14$		0			0
$15$	−6.38741	+	1.82843i	−1.64922	+	0.472098i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$			6.81801i			1.56416i	0.623179	+	0.782080i	$0.285841\pi$
$19$			6.81801i			1.56416i	−0.623179	+	0.782080i	$0.714159\pi$
$20$		0			0
$21$	7.86123			1.71546
$22$		0			0
$23$	−7.48331			−1.56038			−0.780189	−	0.625543i	$-0.784877\pi$
$23$	−7.48331			−1.56038			−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	4.24264	−	2.64575i	0.848528	−	0.529150i
$26$		0			0
$27$	8.40401			1.61735
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		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
$35$	−5.68764	+	1.62811i	−0.961387	+	0.275202i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$			21.3137i			3.41292i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−12.5295	+	3.58664i	−1.86779	+	0.534664i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			20.2581i			2.68326i
$58$		0			0
$59$		−	13.9416i		−	1.81504i	−0.420010	−	0.907519i	$-0.637974\pi$
$59$		−	13.9416i		−	1.81504i	0.420010	−	0.907519i	$-0.362026\pi$
$60$		0			0
$61$	11.4230			1.46257			0.731284	−	0.682073i	$-0.238922\pi$
$61$	11.4230			1.46257			0.731284	+	0.682073i	$0.238922\pi$
$62$		0			0
$63$	15.4206			1.94281
$64$		0			0
$65$	−4.41421	−	15.4206i	−0.547516	−	1.91269i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−22.2349			−2.67677
$70$		0			0
$71$			5.65685i			0.671345i	0.941979	+	0.335673i	$0.108964\pi$
$71$			5.65685i			0.671345i	−0.941979	+	0.335673i	$0.891036\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	12.6060	−	7.86123i	1.45562	−	0.907737i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	16.9706i		−	1.90934i	−0.297670	−	0.954669i	$-0.596210\pi$
$79$		−	16.9706i		−	1.90934i	0.297670	−	0.954669i	$-0.403790\pi$
$80$		0			0
$81$	7.48528			0.831698
$82$		0			0
$83$	13.8368			1.51878			0.759391	−	0.650635i	$-0.225497\pi$
$83$	13.8368			1.51878			0.759391	+	0.650635i	$0.225497\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$			18.9787i			1.98951i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.19560	−	14.6569i	−0.430459	−	1.50376i
$96$		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
$101$	6.38589			0.635420			0.317710	−	0.948188i	$-0.397086\pi$
$101$	6.38589			0.635420			0.317710	+	0.948188i	$0.397086\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	−16.8995	+	4.83756i	−1.64922	+	0.472098i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			15.8745i			1.49335i	0.665190	+	0.746674i	$0.268350\pi$
$113$			15.8745i			1.49335i	−0.665190	+	0.746674i	$0.731650\pi$
$114$		0			0
$115$	16.0871	−	4.60500i	1.50013	−	0.429419i
$116$		0			0
$117$			41.8089i			3.86523i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.49240	+	8.29843i	−0.670141	+	0.742234i
$126$		0			0
$127$	22.4499			1.99211			0.996055	−	0.0887357i	$-0.0282826\pi$
$127$	22.4499			1.99211			0.996055	+	0.0887357i	$0.0282826\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			22.5405i			1.96937i	0.174341	+	0.984685i	$0.444221\pi$
$131$			22.5405i			1.96937i	−0.174341	+	0.984685i	$0.555779\pi$
$132$		0			0
$133$			18.0388i			1.56416i
$134$		0			0
$135$	−18.0663	+	5.17157i	−1.55490	+	0.445098i
$136$		0			0
$137$		−	14.9666i		−	1.27869i	−0.768922	−	0.639343i	$-0.779207\pi$
$137$		−	14.9666i		−	1.27869i	0.768922	−	0.639343i	$-0.220793\pi$
$138$		0			0
$139$		−	3.86733i		−	0.328023i	−0.986458	−	0.164012i	$-0.947557\pi$
$139$		−	3.86733i		−	0.328023i	0.986458	−	0.164012i	$-0.0524434\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	20.7989			1.71546
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		−	10.0000i		−	0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$		−	10.0000i		−	0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			0.211161i			0.0168525i	0.999964	+	0.00842626i	$0.00268219\pi$
$157$			0.211161i			0.0168525i	−0.999964	+	0.00842626i	$0.997318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−19.7990			−1.56038
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−38.4558			−2.95814
$170$		0			0
$171$			39.7383i			3.03886i
$172$		0			0
$173$		−	23.9813i		−	1.82326i	−0.411007	−	0.911632i	$-0.634823\pi$
$173$		−	23.9813i		−	1.82326i	0.411007	−	0.911632i	$-0.365177\pi$
$174$		0			0
$175$	11.2250	−	7.00000i	0.848528	−	0.529150i
$176$		0			0
$177$		−	41.4241i		−	3.11363i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	9.94768			0.739405			0.369703	−	0.929150i	$-0.379460\pi$
$181$	9.94768			0.739405			0.369703	+	0.929150i	$0.379460\pi$
$182$		0			0
$183$	33.9408			2.50898
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	22.2349			1.61735
$190$		0			0
$191$		−	22.6274i		−	1.63726i	−0.574320	−	0.818631i	$-0.694733\pi$
$191$		−	22.6274i		−	1.63726i	0.574320	−	0.818631i	$-0.305267\pi$
$192$		0			0
$193$			26.4575i			1.90445i	0.305392	+	0.952227i	$0.401213\pi$
$193$			26.4575i			1.90445i	−0.305392	+	0.952227i	$0.598787\pi$
$194$		0			0
$195$	−13.1158	−	45.8186i	−0.939242	−	3.28114i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−43.6160			−3.03152
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$		0			0
$213$			16.8080i			1.15167i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	24.7279	−	15.4206i	1.64853	−	1.02804i
$226$		0			0
$227$	−29.2029			−1.93826			−0.969132	−	0.246544i	$-0.920705\pi$
$227$	−29.2029			−1.93826			−0.969132	+	0.246544i	$0.920705\pi$
$228$		0			0
$229$	−26.5344			−1.75344			−0.876720	−	0.481000i	$-0.840274\pi$
$229$	−26.5344			−1.75344			−0.876720	+	0.481000i	$0.840274\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	29.9333i		−	1.96099i	−0.196537	−	0.980497i	$-0.562969\pi$
$233$		−	29.9333i		−	1.96099i	0.196537	−	0.980497i	$-0.437031\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	50.4241i		−	3.27540i
$238$		0			0
$239$		−	30.0000i		−	1.94054i	−0.242028	−	0.970269i	$-0.577812\pi$
$239$		−	30.0000i		−	1.94054i	0.242028	−	0.970269i	$-0.422188\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−2.97127			−0.190607
$244$		0			0
$245$	−15.0481	+	4.30759i	−0.961387	+	0.275202i
$246$		0			0
$247$	−48.9075			−3.11191
$248$		0			0
$249$	41.1127			2.60541
$250$		0			0
$251$		−	1.16979i		−	0.0738362i	−0.999318	−	0.0369181i	$-0.988246\pi$
$251$		−	1.16979i		−	0.0738362i	0.999318	−	0.0369181i	$-0.0117541\pi$
$252$		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
$261$		0			0
$262$		0			0
$263$	15.8745			0.978864			0.489432	−	0.872041i	$-0.337204\pi$
$263$	15.8745			0.978864			0.489432	+	0.872041i	$0.337204\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	19.4108			1.18350			0.591749	−	0.806122i	$-0.298438\pi$
$269$	19.4108			1.18350			0.591749	+	0.806122i	$0.298438\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$			56.3908i			3.41292i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$	15.8759			0.943725			0.471863	−	0.881672i	$-0.343582\pi$
$283$	15.8759			0.943725			0.471863	+	0.881672i	$0.343582\pi$
$284$		0			0
$285$	−12.4662	−	43.5494i	−0.738436	−	2.57965i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$			11.6739i			0.681997i	0.940064	+	0.340998i	$0.110765\pi$
$293$			11.6739i			0.681997i	−0.940064	+	0.340998i	$0.889235\pi$
$294$		0			0
$295$	8.57922	+	29.9706i	0.499502	+	1.74495i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	53.6799i		−	3.10439i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	18.9742			1.09004
$304$		0			0
$305$	−24.5563	+	7.02938i	−1.40609	+	0.402501i
$306$		0			0
$307$	12.8172			0.731515			0.365758	−	0.930710i	$-0.380810\pi$
$307$	12.8172			0.731515			0.365758	+	0.930710i	$0.380810\pi$
$308$		0			0
$309$		0			0
$310$		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
$315$	−33.1500	+	9.48935i	−1.86779	+	0.534664i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	18.9787	+	30.4336i	1.05275	+	1.68815i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	26.4575i		−	1.44123i	−0.693334	−	0.720616i	$-0.743859\pi$
$337$		−	26.4575i		−	1.44123i	0.693334	−	0.720616i	$-0.256141\pi$
$338$		0			0
$339$			47.1674i			2.56178i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	18.5203			1.00000
$344$		0			0
$345$	47.7990	−	13.6827i	2.57341	−	0.736652i
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−37.2197			−1.99233			−0.996163	−	0.0875167i	$-0.972107\pi$
$349$	−37.2197			−1.99233			−0.996163	+	0.0875167i	$0.972107\pi$
$350$		0			0
$351$			60.2843i			3.21774i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−3.48106	−	12.1607i	−0.184755	−	0.645422i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.00000i			0.316668i	0.987386	+	0.158334i	$0.0506123\pi$
$359$			6.00000i			0.316668i	−0.987386	+	0.158334i	$0.949388\pi$
$360$		0			0
$361$	−27.4853			−1.44659
$362$		0			0
$363$	−32.6839			−1.71546
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−22.2619	+	24.6569i	−1.14960	+	1.27327i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	66.7048			3.41739
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			66.9738i			3.37838i
$394$		0			0
$395$	10.4432	+	36.4821i	0.525453	+	1.83561i
$396$		0			0
$397$			17.6164i			0.884144i	0.896980	+	0.442072i	$0.145756\pi$
$397$			17.6164i			0.884144i	−0.896980	+	0.442072i	$0.854244\pi$
$398$		0			0
$399$			53.5980i			2.68326i
$400$		0			0
$401$	36.7696			1.83618			0.918092	−	0.396368i	$-0.129729\pi$
$401$	36.7696			1.83618			0.918092	+	0.396368i	$0.129729\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−16.0913	+	4.60621i	−0.799583	+	0.228885i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		−	44.4699i		−	2.19354i
$412$		0			0
$413$		−	36.8859i		−	1.81504i
$414$		0			0
$415$	−29.7452	+	8.51472i	−1.46014	+	0.417971i
$416$		0			0
$417$		−	11.4909i		−	0.562711i
$418$		0			0
$419$		−	18.3676i		−	0.897316i	−0.893704	−	0.448658i	$-0.851902\pi$
$419$		−	18.3676i		−	0.897316i	0.893704	−	0.448658i	$-0.148098\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	30.2225			1.46257
$428$		0			0
$429$		0			0
$430$		0			0
$431$			18.0000i			0.867029i	0.901146	+	0.433515i	$0.142727\pi$
$431$			18.0000i			0.867029i	−0.901146	+	0.433515i	$0.857273\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	51.0213i		−	2.44068i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	40.7990			1.94281
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	29.7127i		−	1.39602i
$454$		0			0
$455$	−11.6789	−	40.7990i	−0.547516	−	1.91269i
$456$		0			0
$457$			5.29150i			0.247526i	0.992312	+	0.123763i	$0.0394963\pi$
$457$			5.29150i			0.247526i	−0.992312	+	0.123763i	$0.960504\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.6701			−1.19558			−0.597789	−	0.801654i	$-0.703954\pi$
$461$	−25.6701			−1.19558			−0.597789	+	0.801654i	$0.703954\pi$
$462$		0			0
$463$	26.4575			1.22958			0.614792	−	0.788689i	$-0.289240\pi$
$463$	26.4575			1.22958			0.614792	+	0.788689i	$0.289240\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.2212			−0.982000			−0.491000	−	0.871160i	$-0.663368\pi$
$467$	−21.2212			−0.982000			−0.491000	+	0.871160i	$0.663368\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$			0.627417i			0.0289098i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	18.0388	+	28.9264i	0.827675	+	1.32723i
$476$		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
$483$	−58.8281			−2.67677
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.4499			1.01730			0.508652	−	0.860972i	$-0.330144\pi$
$487$	22.4499			1.01730			0.508652	+	0.860972i	$0.330144\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			14.9666i			0.671345i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−13.7279	+	3.92969i	−0.610885	+	0.174869i
$506$		0			0
$507$	−114.263			−5.07458
$508$		0			0
$509$	8.72547			0.386750			0.193375	−	0.981125i	$-0.438057\pi$
$509$	8.72547			0.386750			0.193375	+	0.981125i	$0.438057\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			57.2987i			2.52980i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	71.2548i		−	3.12774i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	9.33612			0.408240			0.204120	−	0.978946i	$-0.434567\pi$
$523$	9.33612			0.408240			0.204120	+	0.978946i	$0.434567\pi$
$524$		0			0
$525$	33.3524	−	20.7989i	1.45562	−	0.907737i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	33.0000			1.43478
$530$		0			0
$531$		−	81.2575i		−	3.52627i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	29.5572			1.26842
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	66.5782			2.84149
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	44.8999i		−	1.90934i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−39.0488			−1.64571			−0.822855	−	0.568251i	$-0.807620\pi$
$563$	−39.0488			−1.64571			−0.822855	+	0.568251i	$0.807620\pi$
$564$		0			0
$565$	−9.76869	−	34.1258i	−0.410972	−	1.43568i
$566$		0			0
$567$	19.8042			0.831698
$568$		0			0
$569$	−2.82843			−0.118574			−0.0592869	−	0.998241i	$-0.518883\pi$
$569$	−2.82843			−0.118574			−0.0592869	+	0.998241i	$0.518883\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$		−	67.2321i		−	2.80866i
$574$		0			0
$575$	−31.7490	+	19.7990i	−1.32403	+	0.825675i
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		0			0
$579$			78.6123i			3.26702i
$580$		0			0
$581$	36.6086			1.51878
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−25.7279	−	89.8777i	−1.06372	−	3.71598i
$586$		0			0
$587$	3.39359			0.140068			0.0700342	−	0.997545i	$-0.477689\pi$
$587$	3.39359			0.140068			0.0700342	+	0.997545i	$0.477689\pi$
$588$		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
$599$		−	11.3137i		−	0.462266i	−0.972922	−	0.231133i	$-0.925757\pi$
$599$		−	11.3137i		−	0.462266i	0.972922	−	0.231133i	$-0.0742432\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	23.6470	−	6.76906i	0.961387	−	0.275202i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			47.6235i			1.91725i	0.284670	+	0.958625i	$0.408116\pi$
$617$			47.6235i			1.91725i	−0.284670	+	0.958625i	$0.591884\pi$
$618$		0			0
$619$			11.2440i			0.451936i	0.974135	+	0.225968i	$0.0725544\pi$
$619$			11.2440i			0.451936i	−0.974135	+	0.225968i	$0.927446\pi$
$620$		0			0
$621$	−62.8899			−2.52368
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11.0000	−	22.4499i	0.440000	−	0.897998i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	33.9411i		−	1.35117i	−0.737280	−	0.675587i	$-0.763890\pi$
$631$		−	33.9411i		−	1.35117i	0.737280	−	0.675587i	$-0.236110\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−48.2612	+	13.8150i	−1.91519	+	0.548232i
$636$		0			0
$637$			50.2129i			1.98951i
$638$		0			0
$639$			32.9706i			1.30430i
$640$		0			0
$641$	−48.0833			−1.89917			−0.949587	−	0.313503i	$-0.898498\pi$
$641$	−48.0833			−1.89917			−0.949587	+	0.313503i	$0.898498\pi$
$642$		0			0
$643$	44.9913			1.77428			0.887142	−	0.461496i	$-0.152687\pi$
$643$	44.9913			1.77428			0.887142	+	0.461496i	$0.152687\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	−13.8707	−	48.4558i	−0.541974	−	1.89333i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−50.8557			−1.97806			−0.989030	−	0.147717i	$-0.952807\pi$
$661$	−50.8557			−1.97806			−0.989030	+	0.147717i	$0.952807\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−11.1005	−	38.7784i	−0.430459	−	1.50376i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	44.8999i		−	1.73076i	−0.501113	−	0.865382i	$-0.667076\pi$
$673$		−	44.8999i		−	1.73076i	0.501113	−	0.865382i	$-0.332924\pi$
$674$		0			0
$675$	35.6552	−	22.2349i	1.37237	−	0.855823i
$676$		0			0
$677$			29.5015i			1.13384i	0.823775	+	0.566918i	$0.191864\pi$
$677$			29.5015i			1.13384i	−0.823775	+	0.566918i	$0.808136\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−86.7696			−3.32502
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	9.21001	+	32.1741i	0.351896	+	1.22931i
$686$		0			0
$687$	−78.8407			−3.00796
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	40.3494i		−	1.53496i	−0.641071	−	0.767482i	$-0.721510\pi$
$691$		−	40.3494i		−	1.53496i	0.641071	−	0.767482i	$-0.278490\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.37984	+	8.31371i	0.0902725	+	0.315357i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	88.9397i		−	3.36401i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.8955			0.635420
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		−	98.9117i		−	3.70948i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	89.1380i		−	3.32892i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−31.2843			−1.15868
$730$		0			0
$731$		0			0
$732$		0			0
$733$			53.6940i			1.98323i	0.129220	+	0.991616i	$0.458753\pi$
$733$			53.6940i			1.98323i	−0.129220	+	0.991616i	$0.541247\pi$
$734$		0			0
$735$	−44.7119	+	12.7990i	−1.64922	+	0.472098i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$	−145.317			−5.33836
$742$		0			0
$743$	7.48331			0.274536			0.137268	−	0.990534i	$-0.456168\pi$
$743$	7.48331			0.274536			0.137268	+	0.990534i	$0.456168\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	80.6465			2.95070
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	50.0000i		−	1.82453i	−0.409605	−	0.912263i	$-0.634333\pi$
$751$		−	50.0000i		−	1.82453i	0.409605	−	0.912263i	$-0.365667\pi$
$752$		0			0
$753$		−	3.47575i		−	0.126663i
$754$		0			0
$755$	6.15370	+	21.4973i	0.223956	+	0.782365i
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	100.007			3.61103
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			49.1933i			1.76936i	0.466198	+	0.884681i	$0.345624\pi$
$773$			49.1933i			1.76936i	−0.466198	+	0.884681i	$0.654376\pi$
$774$		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
$783$		0			0
$784$		0			0
$785$	−0.129942	−	0.453939i	−0.00463784	−	0.0162018i
$786$		0			0
$787$	−8.49148			−0.302688			−0.151344	−	0.988481i	$-0.548360\pi$
$787$	−8.49148			−0.302688			−0.151344	+	0.988481i	$0.548360\pi$
$788$		0			0
$789$	47.1674			1.67920
$790$		0			0
$791$			42.0000i			1.49335i
$792$		0			0
$793$			81.9404i			2.90979i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	33.8272i		−	1.19822i	−0.800666	−	0.599111i	$-0.795521\pi$
$797$		−	33.8272i		−	1.19822i	0.800666	−	0.599111i	$-0.204479\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	42.5624	−	12.1837i	1.50013	−	0.429419i
$806$		0			0
$807$	57.6747			2.03025
$808$		0			0
$809$	−31.1127			−1.09386			−0.546932	−	0.837177i	$-0.684204\pi$
$809$	−31.1127			−1.09386			−0.546932	+	0.837177i	$0.684204\pi$
$810$		0			0
$811$		−	13.0773i		−	0.459208i	−0.973284	−	0.229604i	$-0.926257\pi$
$811$		−	13.0773i		−	0.459208i	0.973284	−	0.229604i	$-0.0737430\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$			110.616i			3.86523i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	26.4575			0.922251			0.461125	−	0.887335i	$-0.347446\pi$
$823$	26.4575			0.922251			0.461125	+	0.887335i	$0.347446\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−41.6457			−1.44642			−0.723208	−	0.690630i	$-0.757333\pi$
$829$	−41.6457			−1.44642			−0.723208	+	0.690630i	$0.757333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		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
$841$	29.0000			1.00000
$842$		0			0
$843$	89.1380			3.07008
$844$		0			0
$845$	82.6695	−	23.6646i	2.84392	−	0.814085i
$846$		0			0
$847$	−29.1033			−1.00000
$848$		0			0
$849$	47.1716			1.61892
$850$		0			0
$851$		0			0
$852$		0			0
$853$			35.4440i			1.21358i	0.794862	+	0.606790i	$0.207543\pi$
$853$			35.4440i			1.21358i	−0.794862	+	0.606790i	$0.792457\pi$
$854$		0			0
$855$	−24.4537	−	85.4264i	−0.836300	−	2.92152i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		−	53.9854i		−	1.84196i	−0.389612	−	0.920979i	$-0.627391\pi$
$859$		−	53.9854i		−	1.84196i	0.389612	−	0.920979i	$-0.372609\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.8745			0.540375			0.270187	−	0.962808i	$-0.412914\pi$
$863$	15.8745			0.540375			0.270187	+	0.962808i	$0.412914\pi$
$864$		0			0
$865$	14.7574	+	51.5532i	0.501765	+	1.75286i
$866$		0			0
$867$	−50.5115			−1.71546
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.8230	+	21.9556i	−0.670141	+	0.742234i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$			34.6863i			1.16994i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	25.4912	+	89.0505i	0.856876	+	2.99340i
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	59.3970			1.99211
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	159.497i		−	5.32546i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−21.3848	+	6.12150i	−0.710854	+	0.203485i
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	37.2197			1.23450
$910$		0			0
$911$			30.0000i			0.993944i	0.867766	+	0.496972i	$0.165555\pi$
$911$			30.0000i			0.993944i	−0.867766	+	0.496972i	$0.834445\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−72.9635	+	20.8862i	−2.41210	+	0.690475i
$916$		0			0
$917$			59.6365i			1.96937i
$918$		0			0
$919$			16.9706i			0.559807i	0.960028	+	0.279904i	$0.0903025\pi$
$919$			16.9706i			0.559807i	−0.960028	+	0.279904i	$0.909697\pi$
$920$		0			0
$921$	38.0833			1.25489
$922$		0			0
$923$	−40.5782			−1.33565
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$			47.7261i			1.56416i
$932$		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
$941$	−27.7566			−0.904839			−0.452419	−	0.891805i	$-0.649439\pi$
$941$	−27.7566			−0.904839			−0.452419	+	0.891805i	$0.649439\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−47.7990	+	13.6827i	−1.55490	+	0.445098i
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			29.9333i			0.969633i	0.874616	+	0.484817i	$0.161114\pi$
$953$			29.9333i			0.969633i	−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$	13.9242	+	48.6427i	0.450577	+	1.57404i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	39.5980i		−	1.27869i
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.2811	−	56.8764i	−0.524109	−	1.83092i
$966$		0			0
$967$	−22.4499			−0.721942			−0.360971	−	0.932577i	$-0.617555\pi$
$967$	−22.4499			−0.721942			−0.360971	+	0.932577i	$0.617555\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			44.1643i			1.41730i	0.705560	+	0.708650i	$0.250695\pi$
$971$			44.1643i			1.41730i	−0.705560	+	0.708650i	$0.749305\pi$
$972$		0			0
$973$		−	10.2320i		−	0.328023i
$974$		0			0
$975$	56.3908	+	90.4264i	1.80595	+	2.89596i
$976$		0			0
$977$			59.8665i			1.91530i	0.287936	+	0.957650i	$0.407031\pi$
$977$			59.8665i			1.91530i	−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			50.9117i			1.61726i	0.588315	+	0.808632i	$0.299791\pi$
$991$			50.9117i			1.61726i	−0.588315	+	0.808632i	$0.700209\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			4.11454i			0.130309i	0.997875	+	0.0651544i	$0.0207540\pi$
$997$			4.11454i			0.130309i	−0.997875	+	0.0651544i	$0.979246\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.n.i.1119.14	yes	16
4.3	odd	2	inner	2240.2.n.i.1119.2	yes	16
5.4	even	2	inner	2240.2.n.i.1119.1	✓	16
7.6	odd	2	inner	2240.2.n.i.1119.3	yes	16
8.3	odd	2	inner	2240.2.n.i.1119.15	yes	16
8.5	even	2	inner	2240.2.n.i.1119.3	yes	16
20.19	odd	2	inner	2240.2.n.i.1119.13	yes	16
28.27	even	2	inner	2240.2.n.i.1119.15	yes	16
35.34	odd	2	inner	2240.2.n.i.1119.16	yes	16
40.19	odd	2	inner	2240.2.n.i.1119.4	yes	16
40.29	even	2	inner	2240.2.n.i.1119.16	yes	16
56.13	odd	2	CM	2240.2.n.i.1119.14	yes	16
56.27	even	2	inner	2240.2.n.i.1119.2	yes	16
140.139	even	2	inner	2240.2.n.i.1119.4	yes	16
280.69	odd	2	inner	2240.2.n.i.1119.1	✓	16
280.139	even	2	inner	2240.2.n.i.1119.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.2.n.i.1119.1	✓	16	5.4	even	2	inner
2240.2.n.i.1119.1	✓	16	280.69	odd	2	inner
2240.2.n.i.1119.2	yes	16	4.3	odd	2	inner
2240.2.n.i.1119.2	yes	16	56.27	even	2	inner
2240.2.n.i.1119.3	yes	16	7.6	odd	2	inner
2240.2.n.i.1119.3	yes	16	8.5	even	2	inner
2240.2.n.i.1119.4	yes	16	40.19	odd	2	inner
2240.2.n.i.1119.4	yes	16	140.139	even	2	inner
2240.2.n.i.1119.13	yes	16	20.19	odd	2	inner
2240.2.n.i.1119.13	yes	16	280.139	even	2	inner
2240.2.n.i.1119.14	yes	16	1.1	even	1	trivial
2240.2.n.i.1119.14	yes	16	56.13	odd	2	CM
2240.2.n.i.1119.15	yes	16	8.3	odd	2	inner
2240.2.n.i.1119.15	yes	16	28.27	even	2	inner
2240.2.n.i.1119.16	yes	16	35.34	odd	2	inner
2240.2.n.i.1119.16	yes	16	40.29	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.97127 q^{3} +(-2.14973 + 0.615370i) q^{5} +2.64575 q^{7} +5.82843 q^{9} +O(q^{10})\)	\(q+2.97127 q^{3} +(-2.14973 + 0.615370i) q^{5} +2.64575 q^{7} +5.82843 q^{9} +7.17327i q^{13} +(-6.38741 + 1.82843i) q^{15} +6.81801i q^{19} +7.86123 q^{21} -7.48331 q^{23} +(4.24264 - 2.64575i) q^{25} +8.40401 q^{27} +(-5.68764 + 1.62811i) q^{35} +21.3137i q^{39} +(-12.5295 + 3.58664i) q^{45} +7.00000 q^{49} +20.2581i q^{57} -13.9416i q^{59} +11.4230 q^{61} +15.4206 q^{63} +(-4.41421 - 15.4206i) q^{65} -22.2349 q^{69} +5.65685i q^{71} +(12.6060 - 7.86123i) q^{75} -16.9706i q^{79} +7.48528 q^{81} +13.8368 q^{83} +18.9787i q^{91} +(-4.19560 - 14.6569i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9}+O(q^{10}) \) 16 * q + 48 * q^9	\( 16 q + 48 q^{9} + 112 q^{49} - 48 q^{65} - 16 q^{81}+O(q^{100}) \) 16 * q + 48 * q^9 + 112 * q^49 - 48 * q^65 - 16 * q^81

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